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Question

Prove that n3 - 7n + 3 is divisible by 3 for all n N.

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Solution

Let pn=n3-7n+3 is divisible by 3 nN.Step I: For n=1,p1=13-7×1+3=1-7+3=-3, which is clearly divisible by 3So, it is true for n=1Step II: For n=k,Let pk=k3-7k+3=3m, where m is any integer, be true kN.Step III: For n=k+1,pk+1=k+13-7k+1+3=k3+3k2+3k+1-7k-7+3=k3+3k2-4k-3=k3-7k+3+3k2+3k-6=3m+3k2+k+2 Using step II=3m+k2+k+2=3p, where p is any integerSo, pk+1 is divisible by 3.

Hence, n3 - 7n + 3 is divisible by 3 for all n N.

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