Prove the following by using the principle of mathematical induction for all n ∈ N :
1/(3.5) + 1/(5.7) + 1/(7.9) + .... + 1/[(2n + 1)(2n + 3) = n/[3(2n + 3)]

Solution:

Let P (n) be the given statement

i.e., P (n) : 1/(3.5) + 1/(5.7) + 1/(7.9) + .... + 1/[(2n + 1)(2n + 3) = n/3(2n + 3)

For n = 1,

P (1) : 1/(3.5) = 1/3(2.1 + 3)

1/15 = 1/(3.5)

1/15 = 1/15, which is true.

Assume that P (k) is true for some positive integer k.

P (k) : 1/(3.5) + 1/(5.7) + 1/(7.9) + .... + 1/[(2k + 1)(2k + 3) = k/[3(2k + 3)] .... (1)

We will now prove that P (k + 1) is also true.

Now, we have

1/(3.5) + 1/(5.7) + 1/(7.9) + .... + 1/[2(k + 1) + 1][2(k + 1) + 3]

= [1/(3.5) + 1/(5.7) + 1/(7.9) + .... + 1/[(2k + 1)(2k + 3)] + 1/[(2k + 3)(2k + 5)]

= k/[3(2k + 3)] + 1/[(2k + 3)(2k + 5)] ....[from (1)]

= [k (2k + 5) + 3] / [3(2k + 3)(2k + 5)]

= [2k² + 5k + 3] / [3(2k + 3)(2k + 5)]

= [2k² + 2k + 3k + 3] / [3(2k + 3)(2k + 5)]

= [2k (k + 1) + 3(k +1)] / [3(2k + 3)(2k + 5)]

= [(2k + 3)(k + 1)] / [3(2k + 3)(2k + 5)]

= (k + 1) / [3(2k + 5)]

= (k + 1) / [3(2(k + 1) + 3)]

Thus P (k + 1) is true, whenever P (k) is true.

Hence, from the principle of mathematical induction, the statement P (n) is true for all natural numbers i.e., n ∈ N .

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NCERT Solutions Class 11 Maths Chapter 4 Exercise 4.1 Question 17

Prove the following by using the principle of mathematical induction for all n ∈ N : 1/(3.5) + 1/(5.7) + 1/(7.9) + .... + 1/[(2n + 1)(2n + 3) = n/[3(2n + 3)]

Summary:

We have proved that 1/(3.5) + 1/(5.7) + 1/(7.9) + .... + 1/[(2n + 1)(2n + 3) = n/[3(2n + 3)] by using the principle of mathematical induction for all n ∈ N