Question

Prove that N(N2-1-1) is divisible by 6.​

Answer 1

Answer:

If f(n)=n(n2+5)

f(k+1)−f(k) =(k+1){(k+1)2+5}−k(k2+5)=3k2+3k+1+5=6⋅k(k+1)2+6 which is divisible by 6 as k(k+1) is even

⟹6∣f(k)⟺6∣f(k+1)

If induction is not mandatory,

n(n2+5)=(n−1)n(n+1)Product of Three consecutive integers +6n