Question

Prove that (2n + 7) < (n + 3)², ∀ n ∈ N.

Answer 1

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

P(n) = (2n + 7) < (n + 3)²

For n = 1, P(1) = (2 × 1 + 7) = 9 < (1 + 3)² = 16,
9 < 16

Hence, given statement is true for n = 1, 

i.e., P(1) is true.

Let given statement is true for n = m. 

i.e., P(m) is true.

Then P(m) = (2m + 7) < (m + 3)²

Now, we have to prove that given statement is true is true for n = k + 1 

i.e., P(m + 1) is true.

Hence, the given statement is also true for n = m + 1, 

i.e., P(m + 1) is true.

Hence, by the principle of mathematical inducation 

we can say that the given statement is true for each natural number n ∈ N.

Hence Proved.