NCERT Solutions
Class 11 Maths
Principle of Mathematical Induction
Ex.4.1 Q.15
Prove the following by using the principle of mathematical induction for all n є N:
12 + 32 + 52 + ……………...+ (2n -1)2 = [{n (2n - 1) (2n + 1)} ÷ 3]
Let the given statement be P(n), i.e.,
P(n): 12 + 32 + 52 + ……………...+ (2n -1)2 = {n (2n - 1) (2n + 1)} ÷ 3
For n = 1, we have
P (1): 12 = 1 = {1(2.1 - 1) (2.1 + 1)} ÷ 3 = (1.1.3) ÷ 3 = 1, which is true.
Let P(k) be true for some positive integer k, i.e.,
P(k): 12 + 32 + 52 + ……………...+ (2k -1)2 = {k (2k - 1) (2k + 1)} ÷ 3 …………. (1)
We shall now prove that P (k + 1) is true.
Consider
12 + 32 + 52 + ……………...+ (2k -1)2 + {2(k + 1) - 1}2
= {k (2k - 1) (2k + 1)} ÷ 3 + {2(k + 1) - 1}2
[From equation 1]
= [k (2k - 1) (2k + 1) + 3{2(k + 1) - 1}2] ÷ 3
= [k (2k - 1) (2k + 1) + 3{2k + 2 - 1}2] ÷ 3
= [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, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.