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

Solution:

Let P (n) be the given statement.

i.e, P (n) : (2n + 7) < (n + 3)²

We note that P (n) is true for n = 1,

Since,

P (1) : (2.1 + 7) < (1 + 3)²

9 < 16, which is true.

Assume that P (k) is true for some positive integer k.

i.e., (2k + 7) < (k + 3)² ....(1)

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

Now, we have

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

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

< (k + 3)² + 2 .... [from (1)]

< k² + 6k + 9 + 2

< k² + 6k + 11

Now,

[(k + 1) + 3]² = (k + 4)²

= k² + 8k + 16

Since,

k² + 6k + 11 < k² + 8k + 16

Therefore,

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

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NCERT Solutions Class 11 Maths Chapter 4 Exercise 4.1 Question 24

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

Summary:

A relation f and how it is defined is given. We have proved the given statement by using the principle of mathematical induction for all n.