Prove the following using the principle of mathematical induction for all n ∈ N

(2n + 7) < (n + 3)².

Let the given statement be P(n), as

P(n):(2n + 7) < (n + 3)²

First, we check if it is true for n = 1,

P(1): (2 + 7) < (4)²;

∴ It is true for n = 1.

Now we assume that it is true for some positive integer k, such that

P(k):(2k + 7) < (k + 3)²

We shall prove that P(k + 1)is true,

⇒ (2k + 7) + 2 < (k + 3)² + 2

⇒ 2(k + 1) + 7 < k² + 6k + 11

⇒ 2(k + 1) + 7 < k² + 6k + 11 + (2k + 5)

⇒ 2(k + 1) + 7 < k² + 8k + 16

⇒ 2(k + 1) + 7 < (k + 4)²

We proved that P(k + 1) is true.

Hence by principle of mathematical induction it is true for all n ∈ N.