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

im suppose to use mathematical induction to prove this is true.

for the inductive step I ended up with this:
((k(k+1))/2)^2 + (k+1)^3 = (((k+1)((k+1)+1))/2)^2

so.......how would you simplify this.....thanks in advance!.

Question

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

im suppose to use mathematical induction to prove this is true.

for the inductive step I ended up with this:
((k(k+1))/2)^2 + (k+1)^3 = (((k+1)((k+1)+1))/2)^2

so.......how would you simplify this.....thanks in advance!.

anonymous · Answer

First multiply by 4
k^2(k+1)^2+4(k+1)^3=((k+1)((k+2))^2
Factor (k+1)^2
(k+1)^2(k^2+4k+4)=((k+1)((k+2))^2
(k+1)^2(k+2)^2=((k+1)((k+2))^2
We are done!