text:
CBSE (Commerce) Class 11CBSE
Share

Books Shortlist

# RD Sharma solutions for Class 11 Mathematics chapter 20 - Geometric Progression

## Chapter 20 : Geometric Progression

#### Pages 9 - 10

Q 1.1 | Page 9

Show that one of the following progression is a G.P. Also, find the common ratio in case:

4, −2, 1, −1/2, ...

Q 1.2 | Page 9

Show that one of the following progression is a G.P. Also, find the common ratio in case:

−2/3, −6, −54, ...

Q 1.3 | Page 9

Show that one of the following progression is a G.P. Also, find the common ratio in case:

$a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .$

Q 1.4 | Page 9

Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...

Q 2 | Page 10

Show that the sequence <an>, defined by an = $\frac{2}{3^n}$, n ϵ N is a G.P.

Q 3.1 | Page 10

Find:
the ninth term of the G.P. 1, 4, 16, 64, ...

Q 3.2 | Page 10

Find:

the 10th term of the G.P.

$- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .$

Q 3.3 | Page 10

Find :

the 8th term of the G.P. 0.3, 0.06, 0.012, ...

Q 3.4 | Page 10

Find :

the 12th term of the G.P.

$\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .$

Q 3.5 | Page 10

Find :

nth term of the G.P.

$\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .$

Q 3.6 | Page 10

Find :

the 10th term of the G.P.

$\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .$

Q 4 | Page 10

Find the 4th term from the end of the G.P.

$\frac{2}{27}, \frac{2}{9}, \frac{2}{3}, . . . , 162$
Q 5 | Page 10

Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?

Q 6.1 | Page 10

Which term of the G.P. :

$\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?$

Q 6.2 | Page 10

Which term of the G.P. :

$2, 2\sqrt{2}, 4, . . .\text { is }128 ?$

Q 6.3 | Page 10

Which term of the G.P. :

$\sqrt{3}, 3, 3\sqrt{3}, . . . \text { is } 729 ?$

Q 6.4 | Page 10

Which term of the G.P. :

$\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?$

Q 7 | Page 10

Which term of the progression 18, −12, 8, ... is $\frac{512}{729}$ ?

Q 8 | Page 10

Find the 4th term from the end of the G.P.

$\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}$

Q 9 | Page 10

The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.

Q 10 | Page 10

The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.

Q 11 | Page 10

If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.

Q 12 | Page 10

If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.

Q 13 | Page 10

The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.

Q 14 | Page 10

In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.

Q 15 | Page 10

If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.

Q 16 | Page 10

If $\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}$ (x ≠ 0), then show that abc and d are in G.P.

Q 17 | Page 10

If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is $\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}$.

#### Page 16

Q 1 | Page 16

Find three numbers in G.P. whose sum is 65 and whose product is 3375.

Q 2 | Page 16

Find three numbers in G.P. whose sum is 38 and their product is 1728.

Q 3 | Page 16

The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.

Q 4 | Page 16

The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is $87\frac{1}{2}$ . Find them.

Q 5 | Page 16

The sum of first three terms of a G.P. is $\frac{39}{10}$ and their product is 1. Find the common ratio and the terms.

Q 6 | Page 16

The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.

Q 7 | Page 16

The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.

Q 8 | Page 16

Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.

Q 9 | Page 16

The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.

#### Pages 27 - 29

Q 1.1 | Page 27

Find the sum of the following geometric progression:

2, 6, 18, ... to 7 terms;

Q 1.2 | Page 27

Find the sum of the following geometric progression:

1, 3, 9, 27, ... to 8 terms;

Q 1.3 | Page 27

Find the sum of the following geometric progression:

1, −1/2, 1/4, −1/8, ... to 9 terms;

Q 1.4 | Page 27

Find the sum of the following geometric progression:

(a2 − b2), (a − b), $\left( \frac{a - b}{a + b} \right)$ to n terms;

Q 1.5 | Page 27

Find the sum of the following geometric progression:

4, 2, 1, 1/2 ... to 10 terms.

Q 2.1 | Page 27

Find the sum of the following geometric series:

0.15 + 0.015 + 0.0015 + ... to 8 terms;

Q 2.2 | Page 27

Find the sum of the following geometric series:

$\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };$

Q 2.3 | Page 27

Find the sum of the following geometric series:

$\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };$

Q 2.4 | Page 27

Find the sum of the following geometric series:

(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;

Q 2.5 | Page 27

Find the sum of the following geometric series:

$\frac{3}{5} + \frac{4}{5^2} + \frac{3}{5^3} + \frac{4}{5^4} + . . . \text { to 2n terms };$

Q 2.6 | Page 27

Find the sum of the following geometric series:

$\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .$

Q 2.7 | Page 27

Find the sum of the following geometric series:

1, −a, a2, −a3, ... to n terms (a ≠ 1)

Q 2.8 | Page 27

Find the sum of the following geometric series:

x3, x5, x7, ... to n terms

Q 2.9 | Page 27

Find the sum of the following geometric series:

$\sqrt{7}, \sqrt{21}, 3\sqrt{7}, . . .\text { to n terms }$

Q 3.1 | Page 28

Evaluate the following:

$\sum^{11}_{n = 1} (2 + 3^n )$

Q 3.2 | Page 28

Evaluate the following:

$\sum^n_{k = 1} ( 2^k + 3^{k - 1} )$

Q 3.3 | Page 28

Evaluate the following:

$\sum^{10}_{n = 2} 4^n$

Q 4.1 | Page 28

Find the sum of the following serie:

5 + 55 + 555 + ... to n terms;

Q 4.2 | Page 28

Find the sum of the following series:

7 + 77 + 777 + ... to n terms;

Q 4.3 | Page 28

Find the sum of the following series:

9 + 99 + 999 + ... to n terms;

Q 4.4 | Page 28

Find the sum of the following series:

0.5 + 0.55 + 0.555 + ... to n terms.

Q 4.5 | Page 28

Find the sum of the following series:

0.6 + 0.66 + 0.666 + .... to n terms

Q 5 | Page 28

How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make $\frac{3069}{512}$ ?

Q 6 | Page 28

How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?

Q 7 | Page 28

How many terms of the sequence $\sqrt{3}, 3, 3\sqrt{3},$  ... must be taken to make the sum $39 + 13\sqrt{3}$ ?

Q 8 | Page 28

The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.

Q 9 | Page 28

The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.

Q 10 | Page 28

The ratio of the sum of first three terms is to that of first 6 terms of a G.P. is 125 : 152. Find the common ratio.

Q 11 | Page 28

The 4th and 7th terms of a G.P. are $\frac{1}{27} \text { and } \frac{1}{729}$ respectively. Find the sum of n terms of the G.P.

Q 12 | Page 28

Find the sum :

$\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .$

Q 13 | Page 28

The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.

Q 14 | Page 28

If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that $S_1^2 + S_2^2$ = S1 (S2 + S3).

Q 15 | Page 28

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)thterm is $\frac{1}{r^n}$ .

Q 16 | Page 28

If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.

Q 17 | Page 29

How many terms of the G.P. 3, $\frac{3}{2}, \frac{3}{4}$ ..... are needed to give the sum $\frac{3069}{512}$ ?

Q 18 | Page 29

A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.

Q 19 | Page 29

If S1, S2, ..., Sn are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ..., n respectively, then prove that S1 + S2 + 2S3 + 3S4 + ... (n − 1) Sn = 1n + 2n + 3n + ... + nn.

Q 20 | Page 29

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.

Q 21 | Page 29

Let an be the nth term of the G.P. of positive numbers.

Let $\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,$ such that α ≠ β. Prove that the common ratio of the G.P. is α/β.

Q 22 | Page 29

Find the sum of 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.

#### Pages 39 - 40

Q 1.1 | Page 39

Find the sum of the following serie to infinity:

$1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty$

Q 1.2 | Page 39

Find the sum of the following serie to infinity:

8 +  $4\sqrt{2}$ + 4 + ... ∞

Q 1.3 | Page 39

Find the sum of the following serie to infinity:

2/5 + 3/52 +2/53 + 3/54 + ... ∞.

Q 1.4 | Page 39

Find the sum of the following serie to infinity:

10 − 9 + 8.1 − 7.29 + ... ∞

Q 1.5 | Page 39

Find the sum of the following serie to infinity:

$\frac{1}{3} + \frac{1}{5^2} + \frac{1}{3^3} + \frac{1}{5^4} + \frac{1}{3^5} + \frac{1}{56} + . . . \infty$

Q 2 | Page 39

Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.

Q 3 | Page 40

Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.

Q 4 | Page 40

If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.

Q 5 | Page 40

Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.

Q 6 | Page 40

Express the recurring decimal 0.125125125 ... as a rational number.

Q 7 | Page 40

Find the rational number whose decimal expansion is $0 . 423$.

Q 8.1 | Page 40

Find the rational numbers having the following decimal expansion:

$0 . \overline3$

Q 8.2 | Page 40

Find the rational numbers having the following decimal expansion:

$0 .\overline {231 }$

Q 8.3 | Page 40

Find the rational numbers having the following decimal expansion:

$3 . 5\overline 2$

Q 8.4 | Page 40

Find the rational numbers having the following decimal expansion:

$0 . 6\overline8$

Q 9 | Page 40

One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.

Q 10 | Page 40

Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.

Q 11 | Page 40

The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.

Q 12 | Page 40

Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.

Q 13 | Page 40

If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively

$\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}$

#### Pages 45 - 46

Q 1 | Page 45

If a, b, c are in G.P., prove that log a, log b, log c are in A.P.

Q 2 | Page 45

If a, b, c are in G.P., prove that $\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}$ are in A.P.

Q 3 | Page 45

Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.

Q 4 | Page 45

Three numbers are in A.P. and their sum is 15. If 1, 3, 9 be added to them respectively, they form a G.P. Find the numbers.

Q 5 | Page 45

The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.

Q 6 | Page 45

The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.

Q 7 | Page 46

The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.

Q 8.1 | Page 46

If a, b, c are in G.P., prove that:

a (b2 + c2) = c (a2 + b2)

Q 8.2 | Page 46

If a, b, c are in G.P., prove that:

$a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3$

Q 8.3 | Page 46

If a, b, c are in G.P., prove that:

$\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}$

Q 8.4 | Page 46

If a, b, c are in G.P., prove that:

$\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}$

Q 8.5 | Page 46

If a, b, c are in G.P., prove that:

(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.

Q 9.1 | Page 46

If a, b, c, d are in G.P., prove that:

$\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}$

Q 9.2 | Page 46

If a, b, c, d are in G.P., prove that:

(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2

Q 9.3 | Page 46

If a, b, c, d are in G.P., prove that:

(b + c) (b + d) = (c + a) (c + d)

Q 10.1 | Page 46

If a, b, c are in G.P., prove that the following is also in G.P.:

a2, b2, c2

Q 10.2 | Page 46

If a, b, c are in G.P., prove that the following is also in G.P.:

a3, b3, c3

Q 10.3 | Page 46

If a, b, c are in G.P., prove that the following is also in G.P.:

a2 + b2, ab + bc, b2 + c2

Q 11.1 | Page 46

If a, b, c, d are in G.P., prove that:

(a2 + b2), (b2 + c2), (c2 + d2) are in G.P.

Q 11.2 | Page 46

If a, b, c, d are in G.P., prove that:

(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.

Q 11.3 | Page 46

If a, b, c, d are in G.P., prove that:

$\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .$

Q 11.4 | Page 46

If a, b, c, d are in G.P., prove that:

(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.

Q 12 | Page 46

If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)

Q 13 | Page 46

If a, b, c are in G.P., then prove that:

$\frac{a^2 + ab + b^2}{bc + ca + ab} = \frac{b + a}{c + b}$
Q 14 | Page 46

If the 4th, 10th and 16th terms of a G.P. are x, y and z respectively. Prove that x, y, z are in G.P.

Q 15 | Page 46

If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.

Q 16 | Page 46

If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p − q, q − r, r − s are in G.P.

Q 17 | Page 46

If $\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}$ are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.

Q 18 | Page 46

If xa = xb/2 zb/2 = zc, then prove that $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P.

Q 19 | Page 46

If a, b, c are in A.P., b,c,d are in G.P. and $\frac{1}{c}, \frac{1}{d}, \frac{1}{e}$ are in A.P., prove that a, c,e are in G.P.

Q 20 | Page 46

If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.

Q 21 | Page 46

If a, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.

Q 22 | Page 46

If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.

Q 23 | Page 46

If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that $a^{b - c} b^{c - a} c^{a - b} = 1$

#### Pages 54 - 55

Q 1 | Page 54

Insert 6 geometric means between 27 and  $\frac{1}{81}$ .

Q 2 | Page 54

Insert 5 geometric means between 16 and $\frac{1}{4}$ .

Q 3 | Page 54

Insert 5 geometric means between $\frac{32}{9}\text{and}\frac{81}{2}$ .

Q 4.1 | Page 55

Find the geometric means of the following pairs of number:

2 and 8

Q 4.2 | Page 55

Find the geometric means of the following pairs of number:

a3b and ab3

Q 4.3 | Page 55

Find the geometric means of the following pairs of number:

−8 and −2

Q 5 | Page 55

If a is the G.M. of 2 and $\frac{1}{4}$ , find a.

Q 6 | Page 55

Find the two numbers whose A.M. is 25 and GM is 20.

Q 7 | Page 55

Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.

Q 8 | Page 55

The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio $(3 + 2\sqrt{2}) : (3 - 2\sqrt{2})$ .

Q 9 | Page 55

If AM and GM of roots of a quadratic equation are 8 and 5 respectively, then obtain the quadratic equation.

Q 10 | Page 55

If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.

Q 11 | Page 55

Prove that the product of n geometric means between two quantities is equal to the nth power of a geometric mean of those two quantities.

Q 12 | Page 55

If the A.M. of two positive numbers a and b (a > b) is twice their geometric mean. Prove that:

$a : b = (2 + \sqrt{3}) : (2 - \sqrt{3}) .$

Q 13 | Page 55

If one A.M., A and two geometric means G1 and G2 inserted between any two positive numbers, show that $\frac{G_1^2}{G_2} + \frac{G_2^2}{G_1} = 2A$

#### Page 56

Q 1 | Page 56

If the fifth term of a G.P. is 2, then write the product of its 9 terms.

Q 2 | Page 56

If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.

Q 3 | Page 56

If logxa, ax/2 and logb x are in G.P., then write the value of x.

Q 4 | Page 56

If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is $\frac{9}{2}$, then write its first term and common difference.

Q 5 | Page 56

If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.

Q 6 | Page 56

If A1, A2 be two AM's and G1G2 be two GM's between and b, then find the value of $\frac{A_1 + A_2}{G_1 G_2}$

Q 7 | Page 56

If second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.

Q 8 | Page 56

Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.

Q 9 | Page 56

Write the product of n geometric means between two numbers a and b

Q 10 | Page 56

If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.

#### Pages 57 - 58

Q 1 | Page 57

If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is

(a) 1/10

(b) 1/11

(c) 1/9.

(d) 1/20

Q 2 | Page 57

If the first term of a G.P. a1a2a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is

(a) −2/5

(b) −3/5

(c) 2/5

(d) none of these

Q 3 | Page 57

If abc are in A.P. and xyz are in G.P., then the value of xb − c yc − a za − b is

(a) 0

(b) 1

(c) xyz

(d) xa yb zc

Q 4 | Page 57

The first three of four given numbers are in G.P. and their last three are in A.P. with common difference 6. If first and fourth numbers are equal, then the first number is

(a) 2

(b) 4

(c) 6

(d) 8

Q 5 | Page 57

If abc are in G.P. and a1/b1/y = c1/z, then xyz are in

(a) AP

(b) GP

(c) HP

(d) none of these

Q 6 | Page 57

If S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P2 is equal to

(a) S/R

(b) R/S

(c) (R/S)n

(d) (S/R)n

Q 7 | Page 57

The fractional value of 2.357 is

(a) 2355/1001

(b) 2379/997

(c) 2355/999

(d) none of these

Q 8 | Page 57

If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is

(a) $\frac{p - q}{q - r}$

(b) $\frac{q - r}{p - q}$

(c) pqr

(d) none of these

Q 9 | Page 57

The value of 91/3 . 91/9 . 91/27 ... upto inf, is

(a) 1

(b) 3

(c) 9

(d) none of these

Q 10 | Page 57

The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is

(a) 1/2

(b) 2/3

(c) 1/3

(d) −1/2

Q 11 | Page 57

If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is

(a) 1/3

(b) 2/3

(c) 1/4

(d) 3/4

Q 12 | Page 57

The nth term of a G.P. is 128 and the sum of its n terms  is 225. If its common ratio is 2, then its first term is

(a) 1

(b) 3

(c) 8

(d) none of these

Q 13 | Page 57

If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is

(a) 1/4

(b) 1/2

(c) 2

(d) 4

Q 14 | Page 57

If abc are in G.P. and xy are AM's between ab and b,c respectively, then

(a) $\frac{1}{x} + \frac{1}{y} = 2$

(b) $\frac{1}{x} + \frac{1}{y} = \frac{1}{2}$

(c) $\frac{1}{x} + \frac{1}{y} = \frac{2}{a}$

(d) $\frac{1}{x} + \frac{1}{y} = \frac{2}{b}$

Q 15 | Page 58

If A be one A.M. and pq be two G.M.'s between two numbers, then 2 A is equal to

(a) $\frac{p ^3 + q^3}{pq}$

(b) $\frac{p^3 - q^3}{pq}$

(c) $\frac{p^2 + q^2}{2}$

(d) $\frac{pq}{2}$

Q 16 | Page 58

If pq be two A.M.'s and G be one G.M. between two numbers, then G2

(a) (2p − q) (p −  2q)

(b) (2p − q) (2q − p)

(c) (2p − q) (p + 2q)

(d) none of these

Q 17 | Page 58

If x is positive, the sum to infinity of the series $\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is$

(a) 1/2

(b) 3/4

(c) 1

(d) none of these

Q 18 | Page 58

If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is

(a) 7

(b) 8

(c) 9

(d) 10

Q 19 | Page 58

Given that x > 0, the sum $\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}$ equals

(a)

(b) x + 1

(c) $\frac{x}{2x + 1}$

(d) $\frac{x + 1}{2x + 1}$

Q 20 | Page 58

In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is

(a) $- \frac{4}{5}$

(b) $\frac{1}{5}$

(b) $\frac{1}{5}$

(c) 4

(d) none of these

Q 21 | Page 58

Let x be the A.M. and yz be two G.M.s between two positive numbers. Then, $\frac{y^3 + z^3}{xyz}$  is equal to

(a) 1

(b) 2

(c) $\frac{1}{2}$

(d) none of these

Q 22 | Page 58

The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to

(a) 64

(b) 16

(c) 32

(d) 0

Q 23 | Page 58

The two geometric means between the numbers 1 and 64 are

(a) 1 and 64

(b) 4 and 16

(c) 2 and 16

(d) 8 and 16

(e) 3 and 16

Q 24 | Page 58

In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is

(a) 0

(b) pq

(c) $\sqrt{pq}$

(d) $\frac{1}{2}(p + q)$

Q 25 | Page 58

Mark the correct alternative in the following question:

Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then p2R3 : S3 is equal to

(a) 1 : 1

(b) (Common ratio)n : 1

(c) (First term)2 : (Common ratio)2

(d) None of these

## RD Sharma solutions for Class 11 Mathematics chapter 20 - Geometric Progression

RD Sharma solutions for Class 11 Maths chapter 20 (Geometric Progression) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 11 Mathematics chapter 20 Geometric Progression are Introduction of Sequence and Series, Sum to N Terms of Special Series, Relationship Between A.M. and G.M., Geometric Progression (G. P.), Arithmetic Progression (A.P.), Concept of Series, Concept of Sequences.

Using RD Sharma Class 11 solutions Geometric Progression exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

Get the free view of chapter 20 Geometric Progression Class 11 extra questions for Maths and can use shaalaa.com to keep it handy for your exam preparation

S


text:
# Signs and Abbreviations

Free access

## Signs and Abbreviations

Crossreferences to documents refer first to the chapter in this volume in which the document is published, then to the number of the document within that chapter. Thus document 9.25 refers to Chapter 9, document 25, and document 2.7.8. refers to document 7.8 in Chapter 2

# Ottoman-Southeast Asian Relations (2 vols.)

• Subjects

## Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 142 79 5

format: plain


text:
Home

>

English

>

Class 12

>

Physics

>

Chapter

>

Ray Optics And Optical Instruments

>

A small telescope has an objec...

# A small telescope has an objective lens of focal length 144 cm and an eye-piece of focal length 6.0 cm. What is the magnifying power of the telescope ? What is the separation between the objective and the eye-piece ?

Khareedo DN Pro and dekho sari videos bina kisi ad ki rukaavat ke!

Updated On: 27-06-2022

Text Solution

0.75 cm1.38 cm1.0 m1.5 m

Solution : The separation between the objective and the eyepiece = Length of the telescope tube f = f_(0) + f_(e)
 Here, f_(0) = 144 cm, = 1.44 m, f_(e) = 6.0 cm = 0.06 m
 :. F = 1.44 + 0.06 = 1.5 m


text:
# R Dataset / Package boot / aml

Submitted by pmagunia on March 9, 2018 - 1:06 PM
Attachment Size
179 bytes
Documentation

## Remission Times for Acute Myelogenous Leukaemia

### Description

The aml data frame has 23 rows and 3 columns.

A clinical trial to evaluate the efficacy of maintenance chemotherapy for acute myelogenous leukaemia was conducted by Embury et al. (1977) at Stanford University. After reaching a stage of remission through treatment by chemotherapy, patients were randomized into two groups. The first group received maintenance chemotherapy and the second group did not. The aim of the study was to see if maintenance chemotherapy increased the length of the remission. The data here formed a preliminary analysis which was conducted in October 1974.

### Usage

aml

### Format

This data frame contains the following columns:

time

The length of the complete remission (in weeks).

cens

An indicator of right censoring. 1 indicates that the patient had a relapse and so time is the length of the remission. 0 indicates that the patient had left the study or was still in remission in October 1974, that is the length of remission is right-censored.

group

The group into which the patient was randomized. Group 1 received maintenance chemotherapy, group 2 did not.

### Note

Package survival also has a dataset aml. It is the same data with different names and with group replaced by a factor x.

### Source

The data were obtained from

Miller, R.G. (1981) Survival Analysis. John Wiley.

### References

Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.

Embury, S.H, Elias, L., Heller, P.H., Hood, C.E., Greenberg, P.L. and Schrier, S.L. (1977) Remission maintenance therapy in acute myelogenous leukaemia. Western Journal of Medicine, 126, 267-272.

--

Dataset imported from https://www.r-project.org.


text:
# Math Help - I need urgent urgent hel with this parabola question

1. ## I need urgent urgent hel with this parabola question

he parabola x=y^2+ay+b crosses the parabola y=x^2 at (1,1) MAKING RIGHT ANGLES. WHAT ARE THE VALUES OF A AND B .

any help is greatly appreciated.

2. Originally Posted by shoro
he parabola x=y^2+ay+b crosses the parabola y=x^2 at (1,1) MAKING RIGHT ANGLES. WHAT ARE THE VALUES OF A AND B .

any help is greatly appreciated.
Since (1, 1) lies on $x=y^2+ay+b$ you know that

$a = -b$ .... (1)

To get another equation, I'd calculate the value of the gradient of the curves $x=y^2+ay+b$ and $y = x^2$ at (1, 1) and then set the product of the gradients equal to -1.

Now solve two equations simultaneously for a and b.


text:
# Turaev on Dijkgraaf-Witten for surfaces

Since Ben slept through the first lecture, I guess it falls to me to describe Turaev’s talk. Fortunately it was on a subject near and dear to my heart: quantum invariants of surfaces. There is a well-known invariant of oriented surfaces attached to a finite dimensional semisimple algebra which I’ll take a couple paragraphs to describe.

Let’s pause for a moment to recall some facts about semisimple algebras. Let Tr: A -> k denote the trace of left multiplication. Consider the symmetric billinear form <x, y> = Tr(xy). Much as the nondegeneracy of the Killing form is equivalent to semisimplicity for Lie algebras, the nondegeneracy of this form is equivalent to the semisimplicity of A (at least over C). Dual to this billinear form is a symmetric map p: k -> A (x) A. Similarly we have the map T_3(x,y,z) = Tr(xyz).

Choose a triangulation of your surface. Following Turaev we call a pair (face, edge of that face) a flag. To every flag we assign the algebra A (thought of as a vector space). To each edge we assign the map $p: k \rightarrow A \otimes A$ where the two A’s represent the two flags containing that edge. To each face we assign the map $T_3: A \otimes A \otimes A \rightarrow k$ where the three A’s represent the three flags containing that face ordered using the orientation of the surface. Contracting all these maps gives a map

$k = k^{\text{edges}} \rightarrow A^{\text{flags}} \rightarrow k^{\text{faces}}= k.$

But this map is just given by a scalar! One can easily prove that this construction doesn’t depend on the choice of triangulation. For more details and history see my recent paperlet. This construction can be extended to a TQFT. I’ve given the most elementary discription, however, there’s a much better motivated way of thinking about this which is very nicely explained in the Quantum Gravity Seminar from winter 2001, track 1 weeks 16 and 17.

What is a good choice for the algebra A? One nice option is the group ring of a finite group. There are two bases for the group ring: the basis of grouplike elements and the basis of matrix elements of representations. One can easily compute this invariant in each of these two representations to get a well-known formula due to Mednykh (chi is the Euler characteristic, the sum is over irreps):

$\# G^{\chi(S)-1} \# \mathrm{Hom}(\pi_1(S), G) = \sum \mathrm{dim}(V)^{\chi(S)}.$

(Please check what this formula says for S the sphere or torus, also check it when G is abelian.) See my paperlet for further details and for the nonorientable version of everything.

Turaev’s starting point in his talk and his recent preprint is to consider another algebra A. Let G be a finite group and c a normalized 2-cocycle. We define the twisted group algebra A to be the formal span of elements of G with the multiplication $g_1 * g_2 = c(g_1, g_2) g_1 g_2.$ (Associativity follows from the cocycle condition, and 1 is still a unit by the normalized condition.) Just as before, this algebra has two nice bases: the basis of group elements and the basis of matrix elements in the projective representations of G associated to c. Turaev computes the invariant in these two bases. The computation in the group basis is much more delicate than it was for c=1, but in his paper (although not in the talk) Turaev proved that it coincides with the Dijkgraaf-Witten invariant (which has a formula using only classical algebraic topology). The computation in the latter basis is again easy, and so Turaev produces a formula for the Dijkgraaf-Witten invariants. In particular, using some well-known integrality properties of projective representations Turaev produces an integrality result for the Dijkgraaf-Witten invariants which is completely unexpected from the point of view of classical topology.

I’d like to add one point to this whole story (which came up in my discussions with Chris Schommer-Pries). Why is it that computing these invariants (say in the trivial c case as in my paper) is so easy? It is because both the group algebra C[G] and the matrix algebra M_n(C) are both examples of groupoid algebras! In the latter case the groupoid is the one with n objects and only one morphism between any two objects (we’ll call such examples of groupoids trivializable).

So we start out with some invariant, which a priori depends on G. But suppose we can prove that it depends only on the group algebra of G. Thus we can see that it can’t distinguish say the two nonabelian groups of order 8. This isn’t terribly useful as many groups don’t have any simpler group with the same group algebra. However, if we can associate that invariant to groupoids and it still only depends on the groupoid algebra then we note that any group G has an isomorphic groupoid ring to a very nice groupoid which is the union of trivializable groupoids!

## 11 thoughts on “Turaev on Dijkgraaf-Witten for surfaces”

1. So am I allowed to think of “only depends on the groupoid algebra” as secretly meaning “only depends on the representation theory”?

2. It’s actually weaker than that. The group algebra only tells you the dimensions of the irreps! In particular, any two abelian groups have the same group algebra (after all its a commutative multimatrix algebra, there’s only one of those of any dimension).

3. any two abelian groups have the same group algebra

“The same” or “isomorphic”? Is it possible to tack on extra information that makes the resulting structures non-isomorphic?

4. If you remember the Hopf algebra structure on the group algebra, you can get the group back.

5. Exactly. Now how do we modify that for groupoids?

6. There is a notion of a “quantum groupoid” which is like a Hopf algebra but includes groupoids as an example. However, this concept only works for the functions on a groupoid, not for the groupoid algebra. Perhaps I’ll write a bigger post on this issue sometime later.

7. the groupoid is the one with n objects and only one morphism between any two objects (we’ll call such examples of groupoids weakly-trivial).

Who is “we” here?

The category theorist’s technical term for these beats is “codiscrete categories” (which necessarily are codiscrete groupoids). Sometimes also “pair groupoid” over the objects. Some people even say “chaotic groupoids”, but, personally, I don’t like that at all.

Also personally, I wouldn’t like to call these groupoids “trivial” (weakly or not). They are trivializable — but not even canonically so.

Well, I don’t really care about terminology, as long as I know what is meant. But just thought I’d mention this.

8. I actually made up that term in about 1 second when writing the post because I didn’t know what they’re called. I wanted a term that suggested “equivalent to the trivial groupoid” but not “isomorphic to the trivial groupoid.” Trivializable is definitely better, I’ll change it to that.

9. By the way, I haven’t thought of a proof, but I would be willing to bet that the fact that codiscrete groupoids are trivializable is just another incarnation of the fact that their groupoid algebras (which are matrix algebras, as you mentioned) are Morita equivalent to the trivial algebra (the ground field itself).

10. Yes, that’s more or less correct. But I would say it differently. First of all it’s better to look at the whole 2-category of groupoids. i.e. the objects are groupoinds (viewed as categories) the morphisms are functors and the 2-morphisms are natural transformations.

Btw: (for discrete groupoids) this is equivalent to the 2-category which has groupoids as objects, “left-principle bibundles” as 1-morphism and maps of bibundles as 2-morphisms. For thise who don’t know what this means, I don’t really want to get into it here (sorry). It’s more or less like a principle bundle, but where we use groupoids instead of groups and we have both a left and a right action.

Now there is a functor from this 2-category to the 2-category of algebras, bimodules, and bimodule maps, which associates to each groupoid it’s groupoid algebra. Each bibundle naturally becomes a bimodule, etc.

Thus if a groupoid is “weakly trivial” = equivalent as a category to a trivial groupoid = “trivializable”, then since functors (of 2-categories) map equivalences to equivalences, the groupoid algebras will be morita equivalent. Note that the converse is false. The two nonabelian groups of order 8 provide a counter example.

11. (for discrete groupoids) this is equivalent to the 2-category which has groupoids as objects, “left-principle bibundles” as 1-morphism and maps of bibundles as 2-morphisms.

In general, I’d think, you want GrpdBitor: objects are arbitrary groupoids, morphisms are Hilsum-Skandalis morphisms of these (groupoid bitorsors with the right technical conditions) and homomorphisms of these.

I gather that’s what you have in mind, anyway.

Still another alternative would be, I think, to look at the 2-category of stacks presented by these groupoids.

the converse is false.

Once I knew a theorem about which kinds of algebras come from which kind of categories as category algebras. Though it seems I forgot the details. I think finite hereditary algebras all come from categories generated from finite graphs. Something like this.

But I guess there are algebras which are not (Morita equivalent to) category algebras for any category?


text:
## anonymous 5 years ago Find the indicated nth partial sum of the arithmetic sequence. 1.6, 4.2, 6.8, 9.4, ..., n = 50

1. amistre64

+2.6 ?

2. anonymous

The nth term = dn + c. I think I use this to solve but when I do none of my answers match up with my choices

3. anonymous

and yes 2.6 is the common difference (d)

4. amistre64

1.6 + 2.6(n) ?? think I am wrong...

5. anonymous

my answer choices are: A. 2090 B.3395 C.3265 D.3266 E.3264

6. anonymous

would it not be 1.6+2.6(n-1) ?? Remembering that you start at n=1

7. amistre64

math: thats what I thought, but doubted myself....

8. anonymous

Well thats what I was doing but as you can see none of my answer choices match that answer

9. anonymous

that's because you are looking for the sum of the terms up o the 50th term

10. anonymous

ohhhh wow silly me so I have to go back and add them all up.... sorry about that

11. anonymous

the idea of the question i assume is for you to find the formula which does the sumation for you

12. amistre64

can you do that with an integral of the equation?

13. anonymous

i think to find n, you could use 50=n/2 (1.6+(n-1)2.6)

14. anonymous

no because that would find the area under an axis of real numbers whereas you are looking to sum natural numbers

15. anonymous

$s \left[ n \right]=n\2(a \left[ 1 \right]+a \left[ n \right])$

16. anonymous

maybe?

17. anonymous

no I don't think that is right either.....

18. anonymous

no its not, keep trying though

19. anonymous

Just kidding I figured it out the answer is c

20. anonymous

1.6+49(2.6)=129 50/2(1.6+129)=3265

21. anonymous

oh i see what you meant now by your notation! My mistake, well done :)

22. anonymous

do any of you know or understand how to figure out a problem like this: Find the sum of the infinite series: $\sum_{i=1}^{\infty}2(-1/4)^i$

23. anonymous

Thanks It took me a sec to think through it and figure it out lol maybe if I had originally read the question better lol :)

24. anonymous

if i remember correctly there is a formula for the sum of an infinite series, first take 2 out as it is a constant and then apply the formula, (try wikipedia for it because i cannot be sure of my memory). As -1<r<1 you can use the formula where r=-1/4

25. anonymous

Ok thanks


text:
## FANDOM

1,168 Pages

Newton's method is a method for approximating the value of the roots of a function that cannot be solved for algebraically. Given the function f(x) and an estimate value for the root x0, the first approximation is

$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$

The second is

$x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}$

and in general

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

The more times this process is repeated, the better the approximation will be.

For best results, Newton's method should be initiated where $f'(x_0)$ is sufficiently large, $f''(x_0)$ is sufficiently small, and $x_0$ is close to the intended root. This will help avoid certain situations involving unexpected roots, or non-converging situation.

## Example

Suppose we are given the function

$f(x) = e^{3x} + x^3$

We will start with the approximation x0 = -0.5. The first approximation will be

$x_1 = -0.5 - \frac{f(-0.5)}{f'(-0.5)} = -0.5 - \frac{e^{-1.5} + (-0.5)^3}{3e^{-1.5} + 3(-0.5)^2}= -0.569$

The second will be

$x_2 = -0.569 - \frac{f(-0.569)}{f'(-0.569)} = -0.569 - \frac{e^{-1.5} + (-0.569)^3}{3e^{-1.5} + 3(-0.569)^2}= -0.567$

Plugging this into the original equation, we get

$f(-0.567) = e^{3 \cdot -0.567} + (-0.567)^3 = 0.000217$

The more approximations we make, the closer to zero the function will become.

## Damped Newton's Method

To help improve convergence, newton's method may be dampened with a constant α from (0,1].

$x_{n+1} = x_n - \alpha_n\frac{f(x_n)}{f'(x_n)}$

Ideally, the each value of α should have the next iteration get as close to the root as possible. Ony possible method of determing α is the Band-Rose algorithm.[1]

## References

Community content is available under CC-BY-SA unless otherwise noted.


text:
## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$\text{Exact: } \dfrac{ \ln 5 - \ln 2}{\ln 2 + 2 \ln 5}$ $\text{Approximately: } 0.234$
Take the natural logarithm of both sides: $\ln 2^{x+1} = \ln 5^{1-2x}$ Since $\ln M^r = r \ln M$, then: $\ln 2^{x+1} = (x+1) \ln 2 \hspace{20pt} \text{and} \hspace{20pt} \ln 5^{1-2x} = (1-2x) \ln 5$ Thus the equation above is equivalent to $(x+1) \ln 2 =(1-2x) \ln 5$ Distribute to obtain: $x \ln 2 + \ln 2 = \ln 5 -2x \ln 5$ Combine the $x$ terms: $x \ln 2+ 2x \ln 5 = \ln 5 - \ln 2$ Factor out the $x$ (the common factor) then solve for $x$: $x(\ln 2 + 2 \ln 5 ) = \ln 5 - \ln 2\\$ $x = \boxed{\dfrac{ \ln 5 - \ln 2}{\ln 2 + 2 \ln 5} \approx 0.234}$


text:
# How do you identify equations as exponential growth, exponential decay, linear growth or linear decay f(x) = 4 (3/2)^x?

For the expression given we have an exponent, so the result will be exponential. Because the exponent is applied to a positive value ($\frac{3}{2}$) it will be a growth function, increasing as "x" increases.


text:
Environ Eng Res > Volume 20(1); 2015 > Article
Kang, Lee, and Kim: Efficient aerobic denitrification in the treatment of leather industry wastewater containing high nitrogen concentration

### Abstract

To treat leather industry wastewater (LIW) containing high nitrogen concentration, eight aerobic denitrifiers were isolated from sludge existing in an LIW-treatment aeration tank. Among them, one strain named as KH8 had showed the great ability in denitrification under an aerobic condition, and it was identified as Pseudomonas aeruginosa R12. The aerobic denitrification ability of the strain KH8 was almost comparable to its anaerobic denitrification ability. In lab-scale aerobic denitrifications performed in 1-L five-neck flasks for 48 hr, denitrification efficiency was found to be much improved as the strain KH8 held a great majority in the seeded cells. From the nitrogen balance at the cell-combination ratio of 10:1 (the strain KH8 to the other seven isolates) within the seeded cells, the percentage of nitrogen loss during the aerobic denitrification process was estimated to be 58.4, which was presumed to be converted to N2 gas. When these seeded cells with lactose were applied to plant-scale aeration tank for 56 day to treat high-strength nitrogen in LIW, the removal efficiencies of CODCr and TN were achieved to be 97.0% and 89.8%, respectively. Under this treatment, the final water quality of the effluent leaving the treatment plant was good enough to meet the water-quality standards. Consequently, the isolated aerobic denitrifiers could be suitable for the additional requirement of nitrogen removal in a limited aeration-tank capacity. To the best of our knowledge, this is the first report of aerobic denitrifiers applied to plant-scale LIW treatment.

### 1. Introduction

Nitrogenous substances remaining in the industrial and domestic wastewater have attracted attention because of the role of nitrogen in eutrophication of receiving waters. Biological nitrification-denitrification is known to be one of the most economical processes for nitrogen removal [1]. Aerobic nitrification and anaerobic denitrification are generally consisted of this process because the nitrogenous substances existing in wastewater are mostly in the form of ammonium ion. It has commonly been accepted that these two biochemical processes are separately conducted for the sake of their different requirements for the reaction parameters, such as dissolve oxygen, substrate sources and retention time [2]. Nowadays, however, a host of reports dealing with a potential way to combine them into an integrated one has been informed for savings in oxygen for nitrification and carbon requirements for denitrification: Ammonium oxidation [3]; aerobic denitrification [4, 5]; nitrification-denitrification [6]; and nitrite nitrification [7]. The utilization of these bacteria is apparently more cost-effective and manageable than the conventional process.
In the nitrogen removal process, various microorganisms participate, and the nitrification process has been considered to be carried out mainly by ammonia- and nitrite-oxidizing bacteria that are obligately aerobic and chemoautotrophic. Nevertheless, a number of heterotrophic microorganisms have been reported to nitrify many types of nitrogen compounds [8]. Thus, understanding the characteristics of bacteria used in the biological nitrification-denitrification process is very important in order to remain high treatment efficiency at all times, and the possible microbial nitrogen conversions should be provided. Currently, aerobic denitrification becomes the center of public interest due to its potential application [9]. Therefore, the isolation and characterization of aerobically denitrifying bacteria have been actively dealt, and the representatively reported bacteria were: Acinetobacter calcoaceticus [10], Pseudomonas stutzei [11, 12], Bacillus subtilis [13], Agrobacterium [14], Marinbacter [15], Pseudomonas mendocina [16], Paracoccus versutus [9], Acinetobacter sp., [17] and Klebsiella pneumoniae [18]. It has been known that these bacteria can not only reduce nitrate under the aerobic condition but also convert ammonium to nitrogen gas via hydroxylamine, nitrite nitrate and nitrous oxide in order. With the advancement of research in the characteristics of these bacteria, the heterotrophic nitrification and aerobic denitrification processes have been clarified to some extent [16]. As a result, it is more feasible that the biological nitrogen removal could be conducted in one aerobic reactor [9].
For feasibility of commercialization, the aerobic denitrifiers have to be indispensably applied to the wastewater treatment plant and examined for their potential. To our best knowledge, no direct application of aerobic denitrifiers to wastewater treatment plant has been performed in spite of many studies on them. Accordingly, the performance of aerobic denitrifier applied to leather industry wastewater (LIW) treatment was tested as a first aim. The LIW is characterized to contain complex pollutants with low biodegradability, and its effluent has to meet the permissible discharge standard for total nitrogen [19]. During the biological nitrogen removal process, the imbalance between carbon and nitrogen concentration in the influent is generally occurred. Hence, high nitrogen concentration contained in the LIW was taken into consideration in this study. In addition, biological properties depend on the size of reactor, although the metabolic patterns remain unchanged [20]. For this reason, candidate aerobic denitrifiers were isolated from the sludge suspended in the aeration tank of LIW treatment plant, the characteristics of their aerobic denitrification in a lab-scale were examined from the nitrogen balance, and their potential ability for N removal in the plant-scale LIW treatment was performed in this study.

### 2.1. Isolation of Potential Aerobic Denitrifiers

The sludge used to isolate the denitrifying bacteria was taken from an aeration tank in the LIW treatment plant located in Busan, Korea. The sludge sample was agitated to obtain homogeneous suspension in sterile 0.2% NaCl. One mL of the suspended liquid was pipetted into a 10-mL tube that contained ‘PYK medium’: 5 g peptone, 3 g yeast extract and 2 g KNO3 in 1 L tap water (pH 7.0). Another 1 mL of the suspended liquid was also pipetted into a 10-mL tube that contained ‘Succinate medium’: 4 g succinate, 1 g (NH4)2SO4, 0.3 g KH2PO4, 0.1 g K2HPO4, 0.2 g Mg SO4·7H2O and 2 g KNO3 in 1 L tap water (pH 7.0). After 3 days incubation at 30°C and 180 rpm, the liquid culture was spread with a platinum loop onto the same media solidified with 1.5% nutrient agar. The separated colonies formed on the agar plates were picked up serially, and a purified isolate was obtained by repeated streaking onto fresh agar plates. Each isolate was maintained on the agar plate at 4°C, and transferred to a fresh agar plate every 2 weeks until use.
The ability of aerobic denitrification for each isolate was tested by the use of the syringe technique under an aerobic condition [21]. First, 1 mL of each isolate was inoculated into a 50-mL glass syringe containing a 20-mL sterile PYK medium. At the beginning, 10 mL of pure O2 was supplied using a Hamilton gastight syringe, after which the syringe reactor was incubated at 30°C and 180 rpm for 72 hr. Oxygen was supplied into the syringe reactor twice more before depleted. To measure the moles of gas produced by denitrification, 20 μL of gas sample was taken periodically using a 50 μL gas-tight syringe and analyzed by gas chromatography (GC). At the same time, a syringe reactor under an anaerobic condition was also tested in parallel. The number of moles of gas produced in the syringe reactor was calculated by the ideal gas law.

### 2.2. Identification of One Potent Isolate

The isolate that produced the largest amount of N2 gas in the syringe test was primarily identified on the basis of colony morphology, Gram reaction, and microscopic examination.
The specific identification of the isolate was performed using 16S-rDNA sequence analysis. Chromosomal DNA of the isolate was extracted from cells grown in the given medium with AccuPrep® Genomic DNA extraction kit (Bioneer, Korea) as the manufacturer’s instructions. PCR amplification of the DNA using the 27F (5″-AGAGTTTGATCCTGGCTCAG-3″) and 1492R (5″-GGTTACCTTGTTACGACTT-3″) primers was performed with a DICE model TP600 PCR thermal cycler (Takara, Japan) as described by Kim et al. [21], and the 16S rDNA genes were determined by the Macrogen Company (Korea). These sequences were compared with GenBank (National Center for Biotechnology Information, USA) entries using the Advanced BLAST similarity search option [22], which is accessible from the homepage of the National Center for Biotechnology Information (http://www.ncbi.nlm.nih.gov/). BioEdit Sequence Alignment Editor version 5.0.9 was used to verify the alignment and remove all positions with gaps before calculating distances with DNAdist programme in PHYLIP (version 3.5c). Phylogenetic tree was gained on the ground of the given sequences and their close relatives using the neighbor joining method with 1,000 bootstrap replicates.

### 2.3. Aerobic Denitrification Experiment in a 1-L Five-Neck Flask

The characteristics of aerobic denitrification by the isolates were investigated in 1-L five-neck flasks. The flask was comprised of five necks, and at each neck, a thermometric sensor, a pH probe, tubing for sampling, oxygen-inlet tubing connected to a membrane filter (0.2-μm pore size), and oxygen-outlet tubing were installed. To avoid contamination from foreign microorganisms, two consecutively connected 1-L flasks containing 10 N NaOH were placed for the discharging gas. This flask set-up was placed in a hot stirring bath system (Eyela, Japan) and maintained at 30 ± 0.2°C. Stirring the medium inside the flask was accomplished with the Variomag Telesystem (H+P Labortechnik AG, Germany) at 500 rpm. Oxygen (1.5 kgf/cm2) was supplied continuously into the flask from an oxygen tank (80% purity), and 10-fold diluted Antifoam 204 was used when the foam occurred severely.
For the preparation of inoculums, each isolate was separately cultivated in a sterile PYK medium until a late-log growth phase. Then, equal amounts (5 mL) harvested from each of eight isolates were combined together and inoculated to the flask containing the PYK medium under an aseptic condition. To examine the effect of one potent isolate on the aerobic denitrification, a flask inoculated with the combined cells at the ratio of 10:1 (one potent isolate to other seven isolates) was cultivated in parallel. During the experiments, the culture broth was sampled periodically from the flask by a peristaltic pump using Tygon tubing, and the changes in the reaction parameters were measured. From these measurements, the percentage of nitrogen converted to N2 gas by the isolated aerobic denitrifiers was calculated as the following formula:
##### (1)
$Initial TN-Final TNInitial TN×100=%N loss$
where the concentration of final TN was the total concentrations of NH4+–N, NO2–N, NO3 –N, organic N and N in biomass at the final stage of aerobic denitrification. The biomass composition was assumed to be C5H7O2N.

### 2.4. Application of Aerobic Denitrifiers to LIW Treatment Plant

To investigate the potential of the isolated aerobic denitrifiers in plant-scale treatment, they were applied to a LIW treatment plant (Busan, Korea) in which effluent quality was difficult to meet the N regulatory standard. The process flow diagram of the LIW treatment plant is shown in Fig. 1. The LIW treatment plant was comprised of collection tank (180 m3), equalization tank (10,472 m3), four relay tanks (758 m3), three clarifiers (6,666 m3), fermentation and synthesis tank (1,769 m3), eight-stage aeration tank (18,572 m3), two sludge thickeners (2,773 m3), two bio-contact tanks for treatment of nitrogen by microalgae (1,144 m3), silica tank for coagulation of colloids (450 m3), carbon tank for adsorption of organic compounds (336 m3), mineral tank for deterioration of organic compounds (453 m3) and industrial water tank (593 m3). The maximal capacity of this plant for LIW treatment was 9,000 m3 per day. The existing treatment for this LIW had been conducted by the use of microbial consortium (Korean patent: 10-1231977) consisting of Bacillus, Paenibacillus, Leuconostoc, Kurthia, Sphingobacteirum, etc. The microbial consortium was seeded daily into equalization tank (0.2%, v/v), fermentation and synthesis tank (0.04%, v/v), aeration tank (0.02%, v/v) and two sludge thickeners (each 0.04%, v/v), respectively. This treatment was set as a control experiment in this study.
Against the control, the experiments using additional input of isolated strains were performed for better N removal of this LIW to meet the N regulatory standard. For the preparation of seed culture, each isolated strain was actively cultivated at 30°C and 180 rpm in a 1-L flask containing the PYK medium, and then cells were collected after centrifugation. The collected cells were combined at a 10:1 ratio of one potent isolate to other seven isolates, based on the result of lab-scale experiment. After 1 day acclimation in the PYK medium at 30°C and 180 rpm under an aerobic condition, the actively grown cells were transferred to a 150 L reactor under an aseptic condition. With addition of 50-mg lactose/L, the seed culture for the plant-scale experiment was further cultivated in a 3-ton tank for 3 days under an aerobic condition. Finally, the proliferated cells prepared in this way were seeded into the aeration tank in two different ways: 1) the proliferated cells of 12 m3 were seeded at the first day and then the cells of 3.6 m3 were seeded daily from the second day to the end (designated as ‘treatment 1’), and 2) The cells were seeded as the above strategy and moreover 0.6-m3 lactose as a carbon source was additionally supplied every 3 days (designated as ‘treatment 2’). The aeration tank was divided into eight stages, and the hydraulic residence time was approximately 42.2 hr (with maximal flow rate of 440 m3/hr). Samples were periodically taken from the aeration tank, and the changes of the reaction parameters were measured according to standard methods for the water pollution. Experiments were carried out for eight weeks.

### 2.5. Analyses

The dry-cell weight (DCW) was determined by weighing the cell pellet after being dried in an oven at 105°C for 12 hr. The cell pellet was prepared by centrifuging a 5 mL sample of broth culture at 5,000 rpm for 10 min and then by decanting the supernatant after washing twice with distilled water. To measure the nitrogen and carbon dioxide gases produced by aerobic denitrifiers in the syringe experiments, 20 μL samples (injection volume) were taken for GC/TCD (Perkin Elmer Instruments) analysis. The columns used were a ‘molecular sieve 13X’ and ‘carboxen 1000’ for nitrogen and carbon dioxide, respectively. In the analyses of both gases, the following conditions were equally applied: the carrier gas was helium at a flow rate of 30 mL/min while the injector and detector temperatures were 100 and 200°C, respectively. However, the oven temperature for nitrogen gas was 40°C, whereas that for carbon dioxide gas was initially 40°C for 3 min and then increased to 170°C at a rate of 30°C/min.
In the aerobic denitrification experiments, the following parameters were analyzed: 5-day biological oxygen demand (BOD5), chemical oxygen demand-dichromate (CODCr), total phosphorus (TP), suspended solids (SS), and total nitrogen (TN), total Kjeldahl nitrogen (TKN), ammonium, nitrite and nitrate. These analyses were carried out according to standard methods for the water pollution [23]. The concentration of dissolved oxygen (DO) and the value of oxidation-reduction potential (ORP) were monitored with an YSI DO probe (Model 58; USA) and Isteck ORP probe (Model 730P; Korea), respectively.

### 3.1. Screening of Potential Aerobic Denitrifiers

By repeated streaking on agar plates, eight strains were purified. The eight isolates were given the names KH1 to KH8, and their characteristics of colonies and morphology were tabulated in Table 1. Each isolate exhibited its own characteristics of colony and morphology distinctively. Among these strains, the strain KH8 exhibited the most prominent ability of N2 gas production in the syringe experiment. For the efficient treatment of nitrous concentration contained in LIW, N2 gas as an end metabolite must be produced from the nitrous compounds at a higher conversion rate. Hence, the denitrification ability of the strain KH8 under an aerobic condition was further examined against that under an anaerobic condition. As shown in Table 2, 72.1-μmole N2 gas was produced anaerobically for 72 hr of cultivation with 28.3-μmole CO2 gas. Under the aerobic condition, N2 production was not active at initial stage. After then, however, N2 was steadily produced, and 64.2-μ mole was produced after 72 hr with a much higher amount of 411.9-μmole CO2. This indicates that the strain KH8 needed some time to adapt to an aerobic condition but its denitrification ability under an aerobic condition was almost comparable to the anaerobic denitrification ability. For this reason, the strain KH8 was identified and characterized in later experiments.

### 3.2. Identification of One Potent Isolate

The species-specific identification of the isolate was performed using 16S rDNA sequence analysis. The 1,418 bp-sized fragment of the 16S rDNA gene of the isolate was amplified and sequenced. Homology searches revealed that the isolate KH8 was closely related to Pseudomonas aeruginosa R12 (DQ073454; 100% similarity) and P. aeruginosa LCD12 (FJ194519; 99% similarity). The phylogenetic tree based on the partial 16S rDNA gene showed the relation of the isolate KH8 and other related strains (Fig. 2). From these distinctive features, the isolate KH8 was designated as P. aeruginosa R12.

### 3.3. Chartacteristics of Isolates in Lab-Scale Aerobic Denitrification

The changes in reaction parameters were examined during the aerobic denitrification using different compositions of isolated strains as seed culture. The characteristics of aerobic denitrification starting with equal amounts of isolated strains under an aerobic condition are shown in Fig. 3(a). The initial DO was measured to be 5.8 mg/L under the continuous supply of oxygen. As the aerobic denitrification started, DO levels decreased to 0.3 mg/L within 2 hr, then fluctuated, and maintained over 0.9 mg/L after 32 hr. This implies that the supplied oxygen did not match the amount of DO consumed by isolates during the active reaction, probably due to the low solubility of oxygen [24]. As DO levels dropped, ORP decreased from 56.7 to −44.0 mV within 1 hr, decreased further to −148.4 mV after 4 hr, then increased and maintained at positive values after 10.5 hr. This indicates that the decrease in ORP value was related to the decrease in DO. The pH was 7.24 at the beginning, and decreased to 7.09 within 1 hr. After this, the pH increased to 8.01 until 5 hr due to alkalinity production by denitrification [25]. After 5 hr, the pH decreased steadily to a final value of 6.20. During the experiment for 48 hr, the concentration of CODCr reduced from 10,695 to 1,632 mg/L, the concentration of TN reduced from 2,355 to 1,767 mg/L, and the concentration of TKN reduced from 1,333 to 588 mg/L. From these data, the removal efficiencies of CODCr, TN and TKN were estimated to be 84.7%, 25.0% and 55.9%, respectively. The resulting CODCr/TN ratio decreased from 8.6 (at the beginning) to 2.0 (at the end). It is known that the C/N ratio may influence the metabolic pathway of organic matter utilization [26]. Accordingly, the decrease in the C/N ratio in a later stage of aerobic denitrification implies that external carbon may be necessary for high efficiency of N removal. Along with these reaction parameters, profiles of NH4+–N, NO2–N and NO3–N concentrations are also shown in Fig. 3(a). The concentration of NH4+–N increased from 73.7 to 354.0 mg/L for first 12 hr, and then decreased slowly to the end. On the other hand, the concentration of NO3–N decreased from 270.7 to 104.6 mg/L for first 12 hr, and then increased slowly to the end. The concentration of NO2–N as an intermediate metabolite in aerobic denitrification maintained low for first 12 hr, then increased to 202.7 mg/L after 24 hr, but decreased to almost zero at the end. From the analyses of all parameters, it was found that active aerobic denitrification took place in the first 12 hr when equal amounts of eight isolates were used as inoculums.
At a cell-combination ratio of 10:1 (the isolate KH8 to the other seven isolates), the characteristics of the aerobic denitrification are shown in Fig. 3(b). From the beginning of the experiment, DO levels maintained below 1 mg/L for the first 5.5 hr even under the continuous supply of oxygen. After then, the DO levels increased and maintained over 2.5 mg/L until the end. Due to active denitrification, the ORP values decreased as DO levels dropped. The ORP decreased to −182 mV after 3 hr, then increased steadily, and maintained at positive values after 11 hr. This was followed by a slight increase to 113.3 mV after 48 hr. The initial pH was 7.24, and decreased slightly to 7.20 within 1 hr. After then, the pH increased steadily to 8.34 until 24 hr. This duration of increase in pH was much longer than that exhibited during the experiment using equal amounts of isolated strains (shown in Fig. 3(a)). This indicates that higher alkalinity production resulted from stronger denitrification by the dominant action of the isolate KH8. After 34 hr, the pH somewhat decreased to a final value of 7.75. In particular, severe foams were generated between 1 hr and 6 hr. As a result, the changes in DO, ORP and pH appeared mostly in the first 12 hr. During the 48-hr reaction, the concentration of CODCr reduced from 7,381 to 1,099 mg/L, the concentration of TN reduced from 3,513 to 1,464 mg/L, and the concentration of TKN reduced 1,471 to 728 mg/L, respectively. From these data, the removal efficiencies of CODCr, TN and TKN were estimated to be 85.1%, 58.3% and 50.5%, respectively. Although there was no significant difference in the CODCr removal, the TN removal efficiency doubled in comparison with that (25.0%) obtained from the experiment using equal amounts of isolated strains (shown in Fig. 3(a)). Thus, this result indicates that the increase in the composition of isolate KH8 in the seeded cells resulted in improvement of N removal efficiency. The initial ratio of CODCr/TN was 5.5, but it decreased to 3.3 after 12 hr. After then, the C/N ratio continued to decrease, and was calculated to be 2.2 at the end. This result indicates that the aerobic denitrification efficiency was influenced by C/N ratio [27]. Along with these reaction parameters, changes in NH4+–N, NO2–N and NO3–N concentrations were also examined. The concentration of NH4+–N increased from 101.1 to 524.0 mg/L for first 12 hr, and then decreased to 144.9 mg/L at the end. On the other hand, the concentrations of NO3–N and NO2–N decreased from 202.3 to 18.0 mg/L and from 24.5 to 0 mg/L for first 12 hr, respectively. After then those concentrations increased slowly to 82.9 and 477.0 mg/L at the end, respectively. Therefore, the final concentration of NO3–N was almost nine times lower, compared with the result shown in Fig. 3(a), indicating that aerobic denitrification was adequately taken place by the dominant isolate KH8.
To examine the characteristics of denitrifying isolates under an aerobic condition, nitrogen balances for their aerobic denitrifications were investigated at different compositions of seeded cells (Table 3). At the cell-combination ratio of 1:7, N loss was estimated to be approximately 32.8% during the 12-hr aerobic denitrification, with the cell concentration of 9 mg/L. This result was presumed that 32.8% of the initial N in the culture medium was converted to N2 gas by the denitrifying activities of isolated strains under an aerobic condition. However, this aerobic denitrification slowed down during the next 36 hr. The N loss taking place in the aerobic denitrification was somewhat improved when the composition of isolate KH8 was increased within the seeded cells (at the cell-combination ratio of 10:1). The percentage of N loss was 42.5 during the first 12 hr, and it was 58.4 during an entire period of this experiment. From these results, the potential of isolate KH8 was found to be obvious in aerobic denitrification. The similar results were found in the previous studies using potential aerobic denitrifiers [10, 13, 20].

### 3.4. Application of Isolates to Plant-Scale LIW Treatment

To examine the potential ability of the isolates for better N removal, plant-scale aerobic denitrification using the isolates was performed in the aeration tank under three different types of treatments (control, treatment 1 and treatment 2). Fig. 4 shows the concentration profiles in the influent and effluent during the entire course of the experimental study. The LIW flowed into this treatment plant had the following characters: pH of 8.5, DO of 1.0 mg/L, CODCr of 3,000 mg/L, BOD5 of 2,500 mg/L, TN of 450 mg/L, NH4+–N of 400 mg/L, TP of 50 mg/L, SS of 3,000 mg/L, and undetectable NO2–N and NO3–N. After the LIW passed through equalization tank, it flowed into fermentation and synthesis tank, and then the eight-stage aeration tank, as shown in Fig. 1. The influent and effluent water quality of the aeration tank under each treatment was tabulated in Table 4. From the control experiment, the influent and effluent concentrations of DO, BOD5, TP and SS were found to be 1.0 and 2.4 mg/L (DO), 1,248 and 71 mg/L (BOD5), 33 and 20 (TP), and 5,174 and 2,976 mg/L (SS), respectively. The increase in SS concentration was due to the return sludge, and this outcoming SS settled down in tertiary clarifier. When the method of treatment 1 (additional input of isolates, as described in the section of 2.4) or treatment 2 (additional input of isolates and C-source, as described in the section of 2.4) was applied, DO levels increased slightly and the removal efficiency of BOD5 also increased slightly. However, the removal efficiencies of TP and SS by these treatments were least different. During the experimental period, the variances in pH, CODCr and TN were obvious among the different types of treatments. The best removal efficiencies of CODCr (97.0%) and TN (89.8%) were achieved by the treatment 2. This TN removal efficiency was much higher than that (51%) obtained from the control experiment. Along with higher TN reduction, a higher pH value (7.34) revealed in the effluent at day 56 under the treatment 2, because more active denitrification occurred under an aerobic condition. In all the treatments, NO2–N was not detected in the effluent at day 56, implying that it converted to N2 by denitrification. The lowest concentrations of NH4+–N (20mg/L) and NO3–N (8.9 mg/L) in the effluent at day 56 were achieved by the treatment 2, in proportion to the TN removal efficiency.
To understand the characteristics of the isolated aerobic denitrifiers in the aeration tank during the plant-scale operation, the major parameters were monitored. The profiles of pH, CODCr and TN values under three different types of treatments are shown in Fig. 4, and those of NH4+–N, NO2–N and NO3–N are shown in Fig. 5. As shown in Fig. 4(a) (in case of control experiment), the pH was 7.20 at the beginning and it ended at 7.03 without significant variance. This result indicates that poor denitrification resulted in pH decrease at the end [28]. The influent concentration of CODCr fluctuated because it was dependent upon daily LIW content. The influent TN was in a range of 290–300 mg/L, and was removed steadily. As it turned out, the removal efficiencies of CODCr and TN were 92.8% and 51%, respectively. The influent C/N ratios were in a range of 3.7–4.0. Hence, the composition of carbons and nitrogens in the influent was not significantly fluctuated. These removal efficiencies were improved when the isolated aerobic denitrifiers were additionally seeded into the aeration tank (in case of treatment 1, as shown in Fig. 4(b)). Those efficiencies were estimated to be 95.6% and 84.2%, respectively. The intense reductions in CODCr and TN started after 14 day, with the increase of pH. The influent pH at 7.18 decreased to 6.78 after 14 day due to strong nitrification. After then, the pH increased steadily to 7.19 due to the alkalinity production by denitrification. These results indicate that the isolated aerobic denitrifiers needed at least 14 day to be established in the aeration tank. The influence of the isolated aerobic denitrifiers on denitrification in the aeration tank was more obvious when these cells were seeded with lactose as carbon source (in case of the treatment 2, as shown in Fig. 4(c)). With addition of a carbon source, the influent C/N ratios (3.9 to 4.2) increased slightly. In this treatment, the pH decreased during the first 14 day, and then it increased by the strong denitrification. The effluent pH revealed 7.34, which was higher by 0.3 than that (7.03) obtained from the control experiment. The similar results could be found in the previous report of heterotrophic nitrification and aerobic denitrification for treatment of high-strength ammonium [29]. By this improved denitrification, the removal efficiencies of CODCr (97.0%) and TN (89.8%) increased further.
As shown in Fig. 5(a), the concentrations of NH4+–N, NO2–N and NO3–N were changed in the control experiment. The influent NH4+–N concentration in a range of 240–255 mg/L converted to NO2–N and NO3–N by nitrification, and final concentration of effluent NH4+–N was 40 mg/L. Although the influent did not contain NO2–N, its concentration in the effluent at day 0 was 10 mg/L due to the effect of nitrification. However, NO2–N was not detected after 40 day. The NO3–N was detected first in the fermentation and synthesis tank. As a result, the initial concentration of NO3–N flowing into the aeration tank was 53 mg/L. The initial NO3–N of 50 mg/L in the effluent increased rapidly to 165 mg/L during the first 14 day, but it did not decrease adequately after 56 day, implying that active denitrification did not occur to offset the nitrification. This reduction of NO3–N after 14 day was obvious in the result of treatment 1 (Fig. 5(b)). The effluent NO3–N of 165 mg/L at 14 day decreased steadily to 20.1 mg/L until 56 day, implying that active denitrification occurred in the aeration tank. Along with NO3–N, the concentration in the effluent NH4+–N decreased from 251 to 35 mg/L in parallel. Similarly, the effluent NO2–N also decreased, and was not detected after 45 day. This trend was more obvious in the result of the treatment 2 (Fig. 5(c)). The concentration of effluent NO3–N increased from 49 to 184 mg/L during the first 14 day, and then decreased rapidly to 8.9 mg/L after 56 day. At the same time the effluent NH4+–N decreased from 247 to 20 mg/L for 56 day. The effluent NO2–N of 9.6 mg/L at the beginning decreased slightly, and was not detected after 47 day. Therefore, more active denitrification by the treatment 2 resulted in the best N removal efficiency among the three different types of treatments. Besides, variances in the values of parameters were not significant after 42 day operation. This result implies that the seeded aerobic denitrifiers for the removal of excess N existing in LIW were stabilized in the aeration tank for six weeks. It is important that operation under a stabilized system is a key to reliable N treatment [30]. Suneethi and Joseph [31] reported that a large-scale anaerobic ammonium oxidation (ANAMMOX) process was stabilized in 130 day after ANAMMOX bacteria were seeded. Compared with this result, the stabilization of the seeded isolates in this study was completed in a shorter time although the LIW contained refractory organic N compounds to some extent [32]. Under the treatment 2, the final water quality in the effluent leaving the treatment plant was as follows: pH 7.3, DO of 3.0 mg/L, CODCr of 30 mg/L, BOD5 of 24 mg/L, TN of 18 mg/L, NH4+–N of 12 mg/L, NO2–N of 0 mg/L, NO3–N of 6 mg/L, TP of 5 mg/L, and SS of 15 mg/L, all of which were below the water-quality standard concentrations. As a result, efficient LIW treatment was achieved by additional input of isolated aerobic denitrifiers with a carbon source. Therefore, the isolated aerobic denitrifiers could be an attractive candidate for application to wastewater treatment plants in which the N removal is difficult after nitrification due to poor denitrification or advanced treatment is required for N under the existing aeration-tank capacity.

### 4. Conclusions

In this study, eight aerobic denitrifiers were newly isolated to treat LIW containing high-strength N efficiently. When one potent isolate (the strain KH8) held a great majority (at the cell-combination ratio of 10:1) within the seeded cells, TN removal was much improved and 58.4% of initial N was presumed to be converted to N2 gas, based on the N balance calculated from the lab-scale aerobic denitrification. When these seeded cells with lactose were applied to plant-scale aeration tank for the treatment of high-strength N contained in LIW, the system could be stabilized after 42 day and the N removal efficiency was approximately 38% higher than that obtained from the existing process. Consequently, the isolated aerobic denitrifiers could be an attractive candidate for application to the efficient removal of high-strength N from wastewater.

### Acknowledgments

This work was supported by Pukyong National University Research Foundation Grant (“Autonomous Ingenuity Research” in Year 2013).

### References

1. Gupta AB, Gupta SKSimultaneous carbon and nitrogen removal from high strength domestic wastewater in an aerobic RBC biofilm. Water Res. 2001;35:1714–1722.

2. Third KA, Gibbs B, Newland M, Cord-Ruwicch RLong-term aeration management for improved N-removal via SND in a sequencing batch reactor. Water Res. 2005;39:3523–3530.

3. Schmidt I, Sliekers O, Schmid M, et alNew concept of microbial treatment processes for the nitrogen removal in wastewater. FEMS Microbiol Rev. 2003;27:481–492.

4. Robertson LA, Kuenen JGThiosphaera pamtotropha gen. nov. sp. nov., a facultatively anaerobic, facultatively autotrophic sulphur bacterium. J Gen Microbiol. 1983;129:2847–2855.

5. Su JJ, Liu BY, Lin J, Yang CPIsolation of an aerobic denitrifying bacterial strain NS-2 from the activated sludge of piggery waste-water treatment systems in Taiwan possessing denitrification under 92% oxygen atmosphere. J Appl Microbiol. 2001;91:853–860.

6. van de Graaf AA, Mulder A, de Bruijn P, Jetten MS, Robertson LA, Kuenen JGAnaerobic oxidation of ammonium is a biologically mediated process. Appl Environ Microbiol. 1995;61:1246–1251.

7. Eum Y, Choi EOptimization of nitrogen removal from piggery waste by nitrite nitrification. Wat Sci Technol. 2002;45:89–96.

8. Focht DD, Verstraete WBiochemical ecology of nitrification and denitrification. Adv Microb Ecol. 1977;1:135–214.

9. Shi Z, Zhang Y, Zhou J, Chen M, Wang XBiological removal of nitrate and ammonium under aerobic atmosphere by Paracoccus versutus LYM. Bioresour Technol. 2013;148:144–148.

10. Zhao B, He YL, Hughes J, Zhang XFHeterotrophic nitrogen removal by a newly isolated. Acinetobacter calcoaceticus HNR. Bioresour Technol. 2010;101:5194–5200.

11. Wan C, Yang X, Lee DJ, Du M, Wan F, Chen CAerobic denitrification by novel isolated strain using NO2−–N as nitrogen source. Bioresour Technol. 2011;102:7244–7248.

12. Zheng M, He D, Ma T, et alReducing NO and N2O emission during aerobic denitrification by newly isolated Pseudomonas stutzeri PCN-1. Bioresour Technol. 2014;162:80–88.

13. Yang XP, Wang SM, Zhang DW, Zhou LXIsolation and nitrogen removal characteristics of an aerobic heterotrophic nitrifying-denitrifying bacterium, Bacillus subtilis A1. Bioresour. Technol. 2011;102:854–862.

14. Chen Q, Ni JAmmonium removal by Agrobacterium sp. LAD9 capable of heterotrophic nitrification-aerobic denitrification. J Biosci Bioeng. 2012;113:619–623.

15. Zheng HY, Liu Y, Gao XY, Ai GM, Miao LL, Liu ZPCharacterization of a marine origin aerobic nitrifying-denitrifying bacterium. J Biosci Bioeng. 2012;114:33–37.

16. Zhu L, Ding W, Feng LJ, Dai X, Xu XYCharacteristics of an aerobic denitrifier that utilizes ammonium and nitrate simultaneously under the oligotrophic niche. Environ Sci Pollut R. 2012;19:3185–3191.

17. Yao S, Ni J, Ma T, Li CHeterotrophic nitrification and aerobic denitrification at low temperature by a newly isolated bacterium, Acinetobacter sp. HA2. Bioresour Technol. 2013;139:80–86.

18. Padhi SK, Tripathy S, Sen R, Mahapatra AS, Mohanty S, Maiti NCharacterisation of heterotrophic nitrifying and aerobic denitrifying Klebsiella pneumoniae CF-S9 strain for bioremediation of wastewater. Int Biodeter Biodegr. 2013;78:67–73.

19. Chung J, Kim J, Kim Y, Hwang YAssessment and selection of best available technology (BAT) for wastewater facilities in the leather tanning and finishing industry. Resour Conserv Recy. 2013;70:32–37.

20. Kim JK, Park KJ, Cho KS, Nam SW, Park TJ, Bajpai RAerobic nitrification–denitrification by heterotrophic Bacillus strains. Bioresour Technol. 2005;96:1897–1906.

21. Kim JB, Cho KS, Jeong SK, Nam SW, Jeong HD, Kim JKIdentification and characterization of a pigment-producing denitrifying bacterium. Biotechnol Bioprocess Eng. 2008;13:217–223.

22. Altschul SF, Madden TL, Schaffer AA, et alGapped BLAST and PSI-BLAST: a new generation of protein database search programs. Nucleic Acids Res. 1997;25:3389–3402.

23. APHA, AWWA. Standard methods for the examination of water and wastewater. 19th edAmerican Water Works Association, Washington: American Public Health Association; 1995.

24. Nielsen J, Villadsen JBioreaction engineering principles. New York: Plenum Press; 1994.

25. Li B, Irvin SThe comparison of alkalinity and ORP as indicators for nitrification and denitrification in a sequencing batch reactor (SBR). Biochem Eng J. 2007;34:248–255.

26. Ruiz G, Jeison D, Chamy RDevelopment of denitrifying and methanogenic activities in USB reactors for the treatment of wastewater: Effect of COD/N ratio. Process Biochem. 2006;41:1338–1342.

27. Kim M, Jeong SY, Yoon SJ, et alAerobic denitrification of Pseudomonas putida AD-21 at different C/N ratios. J Biosci Bioeng. 2008;106:498–502.

28. Pan Y, Ye L, Ni BJ, Yuan ZEffect of pH on N2O reduction and accumulation during denitrification by methanol utilizing denitrifiers. Water Res. 2012;46:4832–4840.

29. Shoda M, Ishikawa YHeterotrophic nitrification and aerobic denitrification of high-strength ammonium in anaerobically digested sludge by Alcaligenes faecalis strain No. 4. J Biosci Bioeng. 2014;117:737–741.

30. Yubo CUI, Tieheng SUN, Lihui ZHAO, Jiang T, Zhang LPerformance of wastewater sludge ecological stabilization. J Environ Sci. 2008;20:385–389.

31. Suneethi S, Joseph KBatch culture enrichment of ANAMMOX populations from anaerobic and aerobic seed cultures. Bioresour Technol. 2011;102:585–591.

32. Lofrano G, Meric S, Zengin GE, Orhon DChemical and biological treatment technologies for leather tannery chemicals and wastewater: A review. Sci Total Environ. 2013;461–462. 265–281.

##### Fig. 1
Process flow diagram of the LIW treatment plant. Arrows indicate the flow of wastewater (→), the flow of sludge (⇢) and the flow of backwash water (→), respectively.
##### Fig. 2
Phylogenetic tree based on a partial 16S rRNA gene of the strain KH8 and other related species.
##### Fig. 3
Changes in reaction parameters during aerobic denitrifications carried out in 1-L 5-neck flasks at different compositions of the seeded cells. The combination ratios of the isolate KH8 to other seven isolates in the seeded cells were 1:7 (a) and 10:1 (b), respectively.
##### Fig. 4
Changes in pH (■), COD (●) and TN (▲) values of plant-scale operation under different types of treatments. (a) Control, (b) Treatment 1, and (c) Treatment 2. Open and closed symbols represent influent and effluent in the aeration tank, respectively.
##### Fig. 5
Change in NH4+–N(▼), NO2–N (◆) and NO3−–N (★) values of plant-scale operation under different types of treatments. (a) Control, (b) Treatment 1, and (c) Treatment 2. Open and closed symbols represent influent and effluent in the aeration tank, respectively.
##### Table 1
Characteristics of Each Isolated Strain
Strain Colony Cell

Color Diameter (mm) Shape Sizea (μm)
KH1 White 1.0–2.0 Short rod L:1.0, W:0.5
KH2 Ivory 0.1 Short rod L:1.0, W:0.5
KH3 Ivory 0.3 Rod L:1.5–2.0, W:0.8–1.0
KH4 Semi-transparent 0.1 Coccus L:1.0, W:0.5
KH5 Ivory 0.5 Rod L:1.5–2.0, W:0.5–0.8
KH6 White 0.5 Rod L:1.5–2.0, W:1.0–1.2
KH7 White 0.5 Coccus L:1, W:0.5
KH8 Ivory 1.0 Rod L:3, W:1

a L and W indicate the length and the width of each cell, respectively.

##### Table 2
Comparison of Characteristics of Denitrification Caused by the Isolate KH8 Between Aerobic and Anaerobic Conditions
Reaction condition N2 production after 8 hr (μmole) N2 production after 72 hr (μmole) CO2 production after 72 hr (μmole)
Anaerobic 37.0±0.8 72.1±1.2 28.3±0.6
Aerobic 10.2±0.4 64.2±0.9 411.9±5.2
##### Table 3
Nitrogen Balances for Aerobic Denitrifications at Different Compositions of Seeded Cells
Cell-combination ratioa Duration (hr) Initial TNb Final N in biomassb N loss (%)

NO3–Nb NO2–Nb NH4+–Nb Org-Nb
1:7 12 2,350 104.6 0 354.0 1,120.28 1.12 32.8
48 735.4 0 105.9 927.46 1.24 24.7
10:1 12 3,510 18.0 0 524.0 1,476.51 1.49 42.5
48 82.9 477.0 144.9 753.96 1.24 58.4

a Combination ratio of the isolate KH8 to other seven isolates within the seeded cells.

b Values are in mg/L.

##### Table 4
Average Values of Reaction Parameters Indicating Water Quality of Influent at day 0 and Effluent at day 56 in the Aeration tank Under Different Types of Treatments
Parameter Controla Treatment 1b Treatment 2c

Influent Effluent Influent Effluent Influent Effluent
pH 7.20 7.03 7.18 7.19 7.26 7.34
DOd 1.0 2.4 1.0 2.9 1.0 3.0
CODd 1,147 83 1,122 49.8 1,154 35
BODd 1,248 71 1,212 42.2 1,203 31
TNd 300 147 303 48 295 30
NH4+–Nd 255 40 251 35 247 20
NO2–Nd 0 0 0 0 0 0
NO3–Nd 53 121.6 49 20.1 49 8.9
TPd 33 20 31 21 32 21
SSd 5,174 2,976 5,153 2,947 5,169 2,931

a Existing treatment by microbial consortium.

b Treatment by isolated aerobic denitrifiers in addition to the existing treatment.

c Treatment by lactose input in addition to the treatment 1.

d Values are in mg/L.

TOOLS
Full text via DOI
E-Mail
Print
Share:
METRICS
 0 Crossref
 0 Scopus
 2,315 View