<s>[INST] Prove that $1^3 + 2^3 + 3^3 +\cdots+ n^3 = \frac14n^4 + \frac12n^3 + \frac14n^2$ I have to prove that this is true using mathematical induction. 
I have this: 
for every $n \in \mathbb N$: $1^3 + 2^3 + 3^3 + ... + n^3 = \frac 14n^4 + \frac 12n^3 + \frac 14n^2$
for $n = 1: 1^3 = 1/4 + 1/2 + 1/4$, hence $P(1)$ is true.
Let $N \in \mathbb N$ be given and assume that $P(N)$ is true, that is $$1^3 + 2^3 + 3^3 + ... + N^3 = \frac 14N^4 + \frac 12N^3 + \frac 14N^2$$
For n = $N$ + 1: 
And now what? I just couldn't solve it.
 [/INST] We have \begin{equation*}
1^3 + 2^3 + 3^3 + ... + N^3 +(N+1)^3= \frac 14N^4 + \frac 12N^3 + \frac 14N^2+(N+1)^3
\end{equation*}
by the induction hypothesis. Also 
\begin{equation*}
\frac{1}{4}(N+1)^4+\frac{1}{2}(N+1)^3+\frac{1}{4}(N+1)^2=\frac 14N^4 + \frac 12N^3 + \frac 14N^2+(N+1)^3
\end{equation*}
</s>