Question

Prove that 1² + 2² + ... + n² > n³/3 ∀ n ∈ N.

Answer 1

Let the given statement is P(n), where n ∈ N

I.e, P(n) = 1² + 2² + .... n² > n³/3, n N

For n = 1, P(n) is true because

P(1) : 1² > 1³/3

1² > 1/3

Let the given statement is true for n = m. 

i.e, P(m) is true.

Then P(m) : 1² + 2² + ... m² > m³/3

Now, we have to prove that given statement is true for P(m + 1) where P(m) is true.

Thus, P(m + 1) is true, when P(m) is true.

Hence, by the principle of mathematical induction,

P(n) is true for n ∈ N.

Hence Proved.