​ For all n ≥ 1 , prove that 1^2 + 2^2 + 3^2 +……….+ n^2 > n^3/3

Question

For all n ≥ 1 , prove that

1² + 2² + 3² +……….+ n² > \(\frac{n^3}{3}\)

Answer 1

Let p(n): 1² + 2² + 3² + n² 

Put n = 1 ⇒ p(1) = 1 > \(\frac{1}{3}\) which is true. 

Assuming that true for p(k)

p(k): 1² + 2² + 3³ +……….+ k² > \(\frac{k^3}{3}\) 

Let p(k + 1): 1² + 2² + 3² +…….+ k² + (k + 1)² > + \(\frac{k^3}{3}\)(k + 1)²

Hence by using the principle of mathematical induction true for all n ∈ N.