Hi

First of all U should consider that :

1+2+3+. . . +n=n(n+1)/2 [1]

(can be simply proven by induction),

all we have to prove is:

1^3+2^3+...+k^3+(k+1)^3=(1+2+...+k+k+1)^2 [2]

According to hypothesis the LHS of the equation above can be written as:

(1+2+...+k)^2+(k+1)^3 ,

which referring to [1] can be reviewed as:

(k(k+1)/2)^2+(k+1)^3=((k+1)(k+2)/2)^2.

which is identical to RHS of [2] and proof is complete.