Let the given statement be P(n), i.e.,

1

3.5

+

1

5.7

+

1

7.9

+ ... +

1

(2n+1)(2n+3)

=

n

3(2n+3)

First we prove that the statement is true for n = 1.
P (1) :

1

3.5

=

1

3(2.1+3)

⇒

1

15

=

1

15

, which is true
Assume P(k) is true for some positive integer k, i.e.,
L.H.S. =

1

3.5

+

1

5.7

+

1

7.9

+ ... +

1

(2k+1)(2k+3)

=

k

3(2k+3)

... (i)
Now we shall prove that P(k + 1) is true.
For this we have to prove that

1

3.5

+

1

5.7

+

1

7.9

+ ... +

1

(2k+1)(2k+3)

+

1

(2(k+1)+1)(2(k+1)+3)

=

(k+1)

3[2(k+1)+3]

L.H.S. =

1

3.5

+

1

5.7

+

1

7.9

+ ... +

1

(2k+1)(2k+3)

+

1

(2(k+1)+1)(2(k+1)+3)

=

k

3(2k+3)

+

1

(2k+3)(2k+5)

[From (i)]
=

1

(2k+3)

[

k

3

+

1

(2k+5)

] =

1

2k+3

[

2k2+5k+3

3(2k+5)

]
=

(2k+3)(k+1)

3(2k+3)(2k+5)

=

(k+1)

3(2k+5)

=

(k+1)

3[2(k+1)+3]

= R.H.S.
Since k ≠ -3/2
Thus P(k + 1) is true, whenever P(k) is true.
Hence, by the principle of mathematical induction P(n) is true ∀ n ∈ N.