Saturday, April 15, 2017

Proof by induction help. I seem to be stuck and my algebra is a little rusty




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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