# Prove that n(n+1)(2n+1) is divisible by 6 for all n?N by using principle of mathematical induction.

Dear student,

Let the given statement be $P\left(n\right)=n\left(n+1\right)\left(2n+1\right)$ which is divisible by 6
For ,  which is true
Assume that $P\left(k\right)$ is true for some positive integers i.e., P(k) is divisible by 6
We shall now prove that P(k + 1) is also true

First term is divisible by 6 as we assumed, and second term is also divisible by 6.
Thus, P(k + 1) is divisible by 6, whenever P(k) is true.
Hence, from the principle of mathematical induction P(n) is true for $n\in N$

Regards

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