Question

Question 24 Prove the following by using the principle of mathematical induction for all n∈N

(2n +7) < (n + 3)^2

Class X1 - Maths -Principle of Mathematical Induction Page 95

Answer 1

(2n + 7) < ( n+3)²
Let P(n):(2n+7)<(n+3)²

step1 :- for n = 1
(2 × 1 + 7) < (1 +3)²
=> 9 < 16
Which is true .

step2 :- for n = K
i.e (2k + 7) < (k +3)²-------(1)

step3 :- for n = k+1
2(k + 1) + 7 = 2k + 2 + 7
( 2k + 7) < (k+3)² from eqn (1)
add both sides, 2
(2k+7) + 2 < (k+3)²+2 = k² + 6k + 9 + 2
= k² + 8k + 16 - 2k -5
= (k+4)² -(2k+5)< (k+4)² = {(k+1)+3}²
hence,
[2(k+1)+7]< [(k+1)+3]²

It means P(k+1) is true when p(k) is true. Hence, the principle of mathematical induction , the statement is true for all natural numbers.