Prove by PMI,n(n+1)(2n+1)is divisible by 6
For n=1
n(n+1)(2n+1) = 6, divisible by 6.
Let the result be true for n=k
Then, k(k+1)(2k+1) is divisible by 6.
So k(k+1)(2k+1) =6m (1)
Now to prove that the result is true for n=k+1
That is to prove, (K+1)(k+2)(2k+3) is divisible by 6.
(K+1)(k+2)(2k+3)=(k+1)k(2k+3)+(k+1)2(2k+3)=(k+1)k(2k+1)+(k+1)k2+(k+1)2(2k+3)
=6m+2(k+1)(k+2k+3) using (1)
=6m+2(k+1)(3k+3)
=6m +6(k+1)(k+1)=6[m+(k+1)2]
So divisible by 6.
Hence, by PMI, the result is true for all n belong to R.