Question 24:

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

Answer

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

{2(k +1) + 7} = (2k + 7) + 2

∴{2(k + 1) + 7} = (2k + 7) + 2 < (k + 3)² + 2                 [using (1)]

2(k + 1) + 7 < k² + 6k + 9 + 2

2(k + 1) + 7 < k² + 6k + 11

Now, k² + 6k + 11 < k² +8k + 16

∴2(K + 1) + 7 < (k + 4)²

2(k + 1) + 7<{(k + 1) + 3}²

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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