NCERT Solutions
Class 11 Maths
Principle of Mathematical Induction
Ex.4.1 Q.2
Prove the following by using the principle of mathematical induction for all n є N:
13 + 23 + 33 + ………...+ n3 = {n (n + 1) ÷ 2}2
Let the given statement be P(n), i.e.,
P(n): 13 + 23 + 33 + ………...+ n3 = {n (n + 1) ÷ 2}2
For n = 1, we have
P (1): 13 =1= {1(1 + 1) ÷ 2}2 = {(1 × 2) ÷ 2}2 = 12 = 1, which is true.
Let P(k) be true for some positive integer k, i.e.,
13 + 23 + 33 + ………...+ k3 = {k (k + 1) ÷ 2}2 ……………... (1)
We shall now prove that P (k + 1) is true.
Consider
13 + 23 + 33 + ... + k3 + (k + 1)3
= (13 + 23 + 33 + .... + k3) + (k + 1)3
= {k (k + 1) ÷ 2}2 + (k + 1)3
= k2(k + 1)2/4 + (k + 1)3
= {k2(k + 1)2 + 4(k + 1)3} ÷ 4
= (k + 1)2{k2 + 4(k + 1)} ÷ 4
= (k + 1)2{k2 + 4k + 4} ÷ 4
= {(k + 1)2 (k + 2)2} ÷ 4
= {(k + 1)2 (k + 1 + 1)2} ÷ 4
= {(k + 1) (k + 1 + 1) ÷ 2}2
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.