Prove the following by using the principle of mathematical induction for all n ∈ N :
1² + 3² + 5² + .... + (2n - 1)² = [n (2n - 1)(2n + 1)]/3

Solution:

Let P (n) be the given statement.

i.e., P (n) : 1² + 3² + 5² + .... + (2n - 1)² = [n (2n - 1)(2n + 1)]/3

For n = 1,

P (1) : 1² = [1 (2.1 - 1)(2.1 + 1)]/3

1 = 1.1.3/3

1 = 1, which is true.

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

P (k) : 1² + 3² + 5² + .... + (2k - 1)² = [k (2k - 1)(2k + 1)]/3 ....(1)

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

Now, we have

1² + 3² + 5² + .... + (2(k + 1) - 1)²

= 1² + 3² + 5² + .... + (2k + 1)²

⇒ [1² + 3² + 5² + .... + (2k - 1)²] + (2k + 1)²

⇒ [k (2k - 1)(2k + 1)]/3 + (2k + 1)² ....[from (1)]

⇒ [k (2k - 1)(2k + 1) + 3(2k + 1)²]/3

⇒ (2k + 1) [k (2k - 1) + 3(2k + 1)]/3

⇒ (2k + 1) [2k² - k + 6k + 3]/3

⇒ (2k + 1) [2k² + 5k + 3]/3

⇒ (2k + 1) [2k² + 2k + 3k + 3]/3

⇒ (2k + 1) [2k (k + 1) + 3(k + 1)]/3

⇒ [(2k + 1)(k + 1)(2k + 3)]/3

⇒ (k + 1) [2 (k + 1) - 1] [2 (k + 1) + 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 15

Prove the following by using the principle of mathematical induction for all n ∈ N : 1² + 3² + 5² + .... + (2n - 1)² = [n (2n - 1)(2n + 1)]/3

Summary:

We have proved that 1² + 3² + 5² + .... + (2n - 1)² = [n (2n - 1)(2n + 1)]/3 by using the principle of mathematical induction for all n ∈ N.