How to determine equation for $\sum_{k=1}^n k^3$ How do you find an algebraic formula for $\sum_{k=1}^n k^3$? I am able to find one for $\sum_{k=1}^n k^2$, but not $k^3$. Any hints would be appreciated.

As you probably have already realized the formula
$$
\sum_{k=1}^n k^3 = \left(\frac{n(n+1)}{2}\right)^2
$$
can be proved using induction.
For $n=1$ you have
$$
1^3 = \left(\frac{1\cdot 2}{2}\right)^2.
$$
Say that the formula holds for $n$ and prove that it holds for $n=1$. So you have/get
$$
\sum_{k=1}^{n+1} k^3 = \left[\sum_{k=1}^n k^3\right] + (n+1)^3 = \frac{n^2(n+1)^2}{2^2} + (n+1)^3.
$$
I will leave you to prove that this right hand side is indeed equal to
$$
\frac{n^2(n+1)^2}{2^2} + (n+1)^3 = \dots = \frac{(n+1)^2(n+2)^2}{2^2}.
$$
