I'm having trouble with induction. Prove $1 + 2^3 + 3^3 + ... + n^3 = \frac{((n^2)(n+1)^2)}4$ I started a new course and I'm expected to know this stuff, and I'm having trouble learning some on my own.
I'm stuck with this problem:
Prove  $1 + 2^3 + 3^3 + ... + n^3 = \frac{((n^2)(n+1)^2)}4$ using induction (the $1+2^3...$ is written in sum notation, although I don't know how to enter that here, sorry).
I started substituting $n \rightarrow n+1$ but I don't know what to do next. Any help would be extremely appreciated.

First, show that this is true for $n=1$:
$\sum\limits_{k=1}^{1}k^3=\frac{1^2(1+1)^2}{4}$
Second, assume that this is true for $n$:
$\sum\limits_{k=1}^{n}k^3=\frac{n^2(n+1)^2}{4}$
Third, prove that this is true for $n+1$:
$\sum\limits_{k=1}^{n+1}k^3=\color{red}{\sum\limits_{k=1}^{n}k^3}+(n+1)^3=\color{red}{\frac{n^2(n+1)^2}{4}}+(n+1)^3=\frac{(n+1)^2(n+2)^2}{4}$

Please note that the assumption is used only in the part marked red.
