Induction involving series with exponents $(1+2+3+\cdots+n)^2=1^3+2^3+3^3+\cdots+n^3$

Question

Does anyone have any hints for proving the following via mathematical induction?

Obviously the base case is easy enough... I'm getting stuck on how these exponents may be manipulated to show $S_k\Rightarrow S_{k+1}$.

Here is the proposition:

Prove that $(1+2+3+\cdots+n)^2=1^3+2^3+3^3+\cdots+n^3$ for every $n\in\mathbb{N}$.

$$\left(\sum_{k=1}^{n+1}k\right)^2=\left(\sum_{k=1}^{n}k\right)^2+2(n+1)\left(\sum_{k=1}^{n}k\right)+(n+1)^2$$ — Bumblebee, Sep 15 '16 at 18:23

score 2 · Accepted Answer · answered Sep 15 '16 at 18:18

2

Hint:

Show that $1+2+...+n=\frac{n(n+1)}{2}$ and $1^3+2^3+...+n^3=(\frac{n(n+1)}{2})^2$ by induction.

answered Sep 15 '16 at 18:18

user2825632

1 Answers1