Prove the following by using the principle of mathematical induction for all n ∈ N :
(1 + 3/1)(1 + 5/4)(1 + 7/9) .... (1 + (2n + 1)/n² = (n + 1)²

Solution:

Let P (n) be the given statement.

i.e., P (n) : (1 + 3/1)(1 + 5/4)(1 + 7/9) .... (1 + (2n + 1)/n² = (n + 1)²

For n = 1,

P (1) : (1 + 3/1) = 4

4 = 4, which is true.

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

i.e., P (k) : (1 + 3/1)(1 + 5/4)(1 + 7/9) .... (1 + (2k + 1)/k² = (k + 1)² ..... (1)

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

Now, we have

[1 + 3/1)(1 + 5/4)(1 + 7/9) .... (1 + [2(k + 1) + 1]/(k + 1)²)

= [1 + 3/1)(1 + 5/4)(1 + 7/9) .... (1 + [2(k + 1)/k²] [1 + (2k + 3)/(k + 1)²]

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

= (k + 1)² + (2k + 3)

= k² + 2k + 1 + 2k + 3

= k² + 4k + 4

= (k + 2)²

= (k + 1 + 1)²

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 13

Prove the following by using the principle of mathematical induction for all n ∈ N : (1 + 3/1)(1 + 5/4)(1 + 7/9) .... (1 + (2n + 1)/n² = (n + 1)²

Summary:

We have proved that (1 + 3/1)(1 + 5/4)(1 + 7/9) .... (1 + (2n + 1)/n² = (n + 1)² by using the principle of mathematical induction for all n ∈ N