How do you "simplify" the sigma sign when it is raised to a power? How do you simplify the following expression:
$$\left(\sum^{n}_{k=1}k \right)^2$$
I am supposed to show that 
$$\left(\sum^{n}_{k=1}k \right)^2 = \sum^{n}_{k=1}k^{3} $$
The problem is I do not really know how to manipulate the sigma sign. I know that I (probably) need to use induction somehow, but the main question is how do you "simplify" the sigma sign when it is raised to a power. Due to the problem itself I know that (most likely); $$\left(\sum^{n}_{k=1}k \right)^2 = \sum^{n}_{k=1}k^{3} $$ so is it possible to simply manipulate the LHS so that it looks like the RHS? 

$$\left(\sum_{k=1}^{n+1}k\right)^2-\left(\sum_{k=1}^{n}k\right)^2=\left(\sum_{k=1}^{n+1}k-\sum_{k=1}^{n}k\right)\left(\sum_{k=1}^{n+1}k+\sum_{k=1}^{n}k\right)$$
$$=(n+1)\left(\frac{(n+1)(n+2)}{2}+\frac{(n+1)(n)}{2}\right)=(n+1)^3$$
I guess you can do the rest now since you have already figured out that you need to use induction.
