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Prove that (13 + 23 + 33+….+n3) = (1+2+3+….+n)2

 

We prove using induction.

Let S(n) suffice for sum of r from r = 1 to r = n

(13 +23 + 33…..n3) = S (n3)

(1+2+3+4….n)2 = ( S n )2

For n=1,n=2,n=3 the result clearly follows.

We assume the result holds for n = k , that is that S (k3) = ( S k )2

We now prove for n = k+1

The following is clear S (k+1)3 = S (k3) + (k+1)3

Secondly

( S (k +1))2 = [1+2+3+4…+k + k+1 ]2 = ( [1+2+3+…k ] + (k+1))2

= [S k + (k+1)]2 = ( S k )2 + 2(k+1) ( S k ) + (k+1)2

= ( S k )2 + 2(k+1)(k)(k+1)/2 + (k+1)2

= ( S k )2 + k(k+1)2 + (k+1)2

= S (k3) + (k+1)(k+1)2 = S (k3) + (k+1)3

Case proven


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