Answer :

Let Assume P(n):12 + 22 + 32 + - - - + n2 =


For n = 1


P(1): 1 =


1=1


= P(n) is true for n = 1


Let P(n) is true for n = k, so


P(k): 12 + 22 + 32 + - - - - - + k2 =


Let’s check for P(n) = k + 1, So


P(k): 12 + 22 + 32 + - - - - - + k2 + (k + 1)2 =


= 12 + 22 + 32 + - - - - - + k2 + (k + 1)2


=


=


=


=


=


=


=


Therefore, P(n) is true for n = k + 1


Hence, P(n) is true for all nN by PMI


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