Prove by mathematical induction that n(n+1) (2n+1) is a multiple of 6 for all n∈N - Maths - Principle of Mathematical Induction

Question

Prove by mathematical induction that n(n+1) (2n+1) is a multiple
of 6 for all n∈N

Subham Mohanty · Answer

When n=1, LHS=1(1+1)(2+)=2*3=6 (which is divisible by 6)
Similarly, when n=k, k(k+1)(2k+) is also divisible by 6 i e
k(k+1)(2k+1) = 6m ------ (1)
Now when n=k+1, we have to prove that (k+1)(k+1+1)[2(k+1) +1} is
a multiple of 6 i e
(k+1)(k+2)(2k+3) is a multiple of 6
Now from (1) - (k+1) = [6m] / (k)(2k+1) --------- (2)
So now LHS = (k+1)(k+2)(2k+3) = { [6m] / (k)(2k+1) } *
(k+2)(2k+3) which is divisible by 6 and hence a multiple of 6
[ HENCE PROVED ]