I'm attempting to solve this problem, and I've reached an impasse. Because I'm attempting to solve for the first value.
$$ \sum_{i=1}^N N^3 = \left( \sum_{i=1}^N i \right)^2 $$
I then get $(N(N+1)/2)^2$ which then expands to $(N^4 + 2N^3 + N^2)/2$.
After this point I'm lost and not sure where to go to continue the problem.
Guide:
try mathematical induction.
Base case: left as exercise.
Suppose $$\sum_{i=1}^{N-1} i^3 = \left( \sum_{i=1}^{N-1} i\right)^2$$
You want to show that
$$\sum_{i=1}^{N} i^3 = \left( \sum_{i=1}^{N} i\right)^2=\left(\frac{N(N+1)}{2} \right)^2$$
Start from LHS,
$$\sum_{i=1}^{N} i^3 = \sum_{i=1}^{N-1} i^3 + N^3=\left( \sum_{i=1}^{N-1} i\right)^2 + N^3=\left(\frac{(N-1)N}{2} \right)^2+N^3$$
Try to show that $\left(\frac{(N-1)N}{2} \right)^2+N^3=\left(\frac{N(N+1)}{2} \right)^2$
btw, there is a typo in your question, the LHS should be $\sum_{i=1}^{N} i^3$