Prove that following by using the principle of mathematical induction for all $$n\in N$$ $$(2n+7)<(n+3)^2$$

Asked by Pragya Singh | 1 year ago |  108

##### Solution :-

We can write the given statement as

P(n): (2n +7) < (n + 3)2

If n = 1 we get

2.1 + 7 = 9 < (1 + 3)2 = 16

Which is true.

Consider P (k) be true for some positive integer k

(2k + 7) < (k + 3)2 … (1)

Now let us prove that P (k + 1) is true.

Here

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

We can write it as

= {2 (k + 1) + 7}

From equation (1) we get

(2k + 7) + 2 < (k + 3)2 + 2

By expanding the terms

2 (k + 1) + 7 < k2 + 6k + 9 + 2

On further calculation

2 (k + 1) + 7 < k2 + 6k + 11

Here k2 + 6k + 11 < k2 + 8k + 16

We can write it as

2 (k + 1) + 7 < (k + 4)2

2 (k + 1) + 7 < {(k + 1) + 3}2

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

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

Answered by Abhisek | 1 year ago

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