Prove the Following by Using the Principle of Mathematical Induction for All N ∈ N (2n +7) < (N + 3)2

Question

Prove the following by using the principle of mathematical induction for all n ∈ N (2n +7) < (n + 3)²

Solution

Let the given statement be P(n), i.e.,

P(n): (2n +7) < (n + 3)²

It can be observed that P(n) is true for n = 1 since 2.1 + 7 = 9 < (1 + 3)² = 16, which is true.

Let P(k) be true for some positive integer k, i.e.,

(2k + 7) < (k + 3)² … (1)

We shall now prove that P(k + 1) is true whenever P(k) is true.

Consider

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

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