<s>[INST] Prove $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ using Induction Prove $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$ using Induction
My proof so far:
Let $P(n)$ be $1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$

Base Case
$P(1):$
LHS = $1^3 = 1$
RHS = $(1)^2 = 1$
Since LHS = RHS, therefore base case holds

Induction Hypothesis
Let $n \in \mathbb{N}$  be arbitrary
Assume $P(n)$ holds

Induction Step
Prove $P(n+1)$ holds:
$$
\begin{align}
& 1^3 + 2^3 + \cdots + \;n^3 + (n+1)^3 \\
= {} & (1 + 2 + \cdots + \; n)^2 + (n+1)^3 \text{ (by Induction Hypothesis)} \\
= {} & (1 + 2 + \cdots + \; n)^2 + (n^3 + 3n^2 + 3n + 1)
\end{align}
$$

This is where I get stuck. I don't know how to prove that my last step is equivalent to:
$$(1 + 2 + \cdots + \;n + (n+1))^2$$
 [/INST] Hint: by the binomial theorem,
$(1+2+\cdots +n +(n+1))^2=(1+2+\cdots +n)^2+(n+1)^2+2(n+1)(1+2+ \cdots n)$.
Now, you can show by induction (or by Gauss' trick) that $1+2+\cdots +n= \dfrac{n(n+1)}{2}$, and hus reach the answer.
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