Mathematical Induction problem

[Fizex]

I'm trying to solve this problem from CH2 of spivak's calculus of which I am self-studying.

Homework Statement

Prove the following by mathematical induction:
$$1^3+...+n^3=(1+...+n)^2$$

Homework Equations

To prove by mathematical induction, you test whether P(1) is true and if P(k) is true then P(k+1) is true.

The Attempt at a Solution

$$1^3+...+n^3=(1+...+n)^2$$
$$1^3+...+n^3+(n+1)^3=(1+...+n)^2+(n+1)^3$$
$$(1+...+n)^2+(n+1)^3=(1+...+n+1)^2$$

From there I have no clue and I've been staring at that for 15 minutes.

 

[Mentallic]

Don't forget that the RHS is

$$(1+2+...+n+(n+1))^2$$

(I'm just making sure that we are on the same page here, since what you wrote can be mistaken as missing that extra n).

Now take the RHS and treat 1+2+...+n as one constant on its own, and expand.

 

[hunt_mat]

Show that it's true for n=1 (or n=2 even)
1^3=1^2, true for n=1, 1^3+2^3=9=(1+2)^2=3^2=2, so true for n=2. Suppose there is a k such that:
$$1^{3}+2^{3}+\cdots +k^{3}=(1+2+\cots +k)^{2}$$
So, we write:
$$(1+2+\cdots +k+(k+1))^{2}=(1+2+\cdots +k)^{2}+(k+1)^{2}+2(k+1)(1+2+\cdots +k)$$
Now you should know a formula for (1+2+...+k), the equation becomes:
$$(1+2+\cdots +k+(k+1))^{2}=1^{3}+2^{3}+\cdots +k^{3}+(k+1)^{2}+2(k+1)(1+2+\cots +k)$$
So you now have to show that:
$$(k+1)^{3}=(k+1)^{2}+2(k+1)(1+2+\cdots +k)$$
How would you go about doing this?