Stuck on a homework question with mathematical induction, I just need some help factoring and am getting stuck.
$\displaystyle \sum_{1 \le j \le n} j^3 = \left[\frac{k(k+1)}{2}\right]^2$
The induction part is: $\displaystyle \left[\frac{k(k+1)}{2}\right]^2
+(k+1)^3$ is where I am having a problem.
If you could give me some hints as to where to go since I keep getting stuck or writing the wrong equation.
I'll get to $\displaystyle \left[{k^2+2k\over2}\right]^2 + 2{(k+1)^3\over2}$
Any push in the right direction will be appreciated.
Answer
$(\frac{k(k+1)}{2})^2+(k+1)^3$
$=\frac{k^2(k+1)^2}{4}+(k+1)(k+1)^2$
$=\frac{(k+1)^2}{4}(k^2+4k+4)$
$=\frac{(k+1)^2}{4}(k+2)^2$
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