n=1, left =1*2=2, right =1*2*3/3 =2,

assume n = k , we have sum of i(i+1) from 1 to k = k(k+1)(k+2)/3k+2)/3

n=k+1 sum of i(i+1) from 1 to k+1 = k(k+1)(k+2)/3 + (k+1)(k+2) = (k+1)（k+2)[k/3+1)

=（k+1)(k+2)(k+3)/3

thus the identify holds for n = k+1,

from principle of mathematical induction, the identity holds for every n>0

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