Question

Question 19 Prove the following by using the principle of mathematical induction for all n∈N: n (n + 1) (n + 5) is a multiple of 3.

Class X1 - Maths -Principle of Mathematical Induction Page 95

Answer 1

n(n+1)(n+5), is multiple of 3.
Let P(n):n(n+1)(n+5) is multiple of 3
step1 :- for n= 1
P(1):1×(1+1)(1+5)=1×2×6
It is divisible by 3.

step2:- for n= k
P(k):k(k+1)(k+5) is divisible by 3
Let 3L = k(k+1)(k+5)
3L = k³ + 6k² + 5k-------(1)

step3:- for n = k+1
P(k):(k+1)(k+2)(k+6)
= {k² + 3k + 2}(k+6)
=6k² + 18k + 12+ k³ + 3k² + 2k
= k³ + 9k² + 20k + 12
From eqn (1)
= (3L-6k²-5k) + 9k² + 20k + 12
= 3L + 3k² + 15k + 12
= 3( L + k² + 5k + 4)
It is divisible by 3. P(k+1) is true when p(k) is true. From the principle of mathematical induction, statement is true.