Question

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

Class X1 - Maths -Principle of Mathematical Induction Page 94

Answer 1

1³ + 2³ + 3³ +.....n³ = [n(n+1)/2]²
Let p(n):1³ + 2³ + 3³ +....n³ = [n(n+1)/2]²

step1:- for n = 1
P(1): 1 = [1(1+1)/2]² = 1
It's true.

step2:- for n = k
P(k): 1³ + 2³ + 3³ +.....k³ = [k(k+1)/2]² --------(1)

step3:- for n = k+1
From eqn (1)
1³ + 2³ + 3³ +......k³ = [k(k+1)/2]²
add (k+1)³ both sides,
1³ + 2³ + 3³ +.....k³ + (k+1)³ = [ k(k+1)/2]² + (k+1)³
= [k²(k+1)² + 4(k+1)³]/4
=[(k+1)²(k² + 4k+4)]/4
= (k+1)²(k+2)²/4
= [ (k+1){(k+1)+1}/2]²
Hence, p(k+1)is true when p(k) is true .from the principle of mathematical induction, statement is true for all natural numbers.