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(n)=nn+12n+1 which is divisible by 6
For n=1,  P1: 1×1+12+1=6,  which is true
Assume that P(k) 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
                  P(k+1)=k+1k+22k+1+1=k+1k+22k+3=k+1k+22k+1+2             =kk+12k+1+2+2k+12k+1+2             =k(k+1)(2k+1)+2k(k+1)+2(k+1)(2k+1)+4(k+1)             =k(k+1)(2k+1)+2(k+1)k+2k+1+2            =k(k+1)(2k+1)+2(k+1)3k+3=k(k+1)(2k+1)+6(k+1)k+1             
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 nN


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