Question

Question 18 Prove the following by using the principle of mathematical induction for all n∈N: 1+2+3+... + n < (2n+1)^2 / 8

Class X1 - Maths -Principle of Mathematical Induction Page 95

Answer 1

1 + 2 + 3 +........n < (2n+1)²/8

Let p(n):1 +2 + 3 + 4 +...n <(2n+1)²/8

step1:- for n = 1
P(1):1 < (2×1+1)²/8 = 1
It is true .

step2:- for n= k
P(k): 1 + 2 + 3 +....k <(2k+1)²/8---(1)

step3:- for n= k+1
P(k+1):1 + 2 + 3+....k+(k+1)<{2(k+1)+1}²/8.

from eqn (1)
1 + 2 + 3 + ...k < (2k+1)²/8
add both sides, (k+1)
1 + 2 + 3 + ..k+(k+1)< (2k+1)²/8 + (k+1)
= {(2k+1)² + 8(k+1)}/8
={4k² + 4k +1 + 8k + 8}/8
= {4k² + 12k+9}/8
= (2k+3)²/8
= [2(k+1)+1 ]²/8
hence,
1 + 2 + 3 +.....k+(k+1)< [2(k+1)+1]²/8
p(k+1) is true , when p(k) is true . from the principle of mathematical induction, statement is true for all natural numbers.