<s>[INST] Is my proof of $x^n-y^n = \ldots$ correct? I am solving problems from Spivak's calculus book's 1st chapter.  Basically Spivak wants me to prove this:
$$x^n-y^n = (x-y)(x^{n-1} + x^{n-2}y + \ldots + xy^{n-2} + y^{n-1})$$
Question 1: Is my proof correct?
Question 2: How to make it perfect?  Any advise?

My proof:
I guess Spivak wants to say:
$$
x^n-y^n = (x-y)\left(\sum_{i=1}^{i=n} x^{n-i}y^{i-1}\right)
$$
Let:
$$
f(x,y,n) = \left(\sum_{i=1}^{i=n} x^{n-i}y^{i-1}\right)
$$
Then we can say (is this sub-proof perfect?):
$$\begin{split}
x^{n+1}-y^{n+1} &= (x-y)\left(\sum_{i=1}^{i=n+1} x^{(n+1)-i}y^{i-1}\right)\\
                &= (x-y)\left(
                        \begin{split}
                            &x^{(n+1)-(n+1)} y^{(n+1)-1}\\
                            &+ \sum_{i=1}^{i=n} x^{(n+1)-i}y^{i-1}
                        \end{split}
                    \right)\\
                &= (x-y)\left(
                        x^{0} y^{n}
                        + \sum_{i=1}^{i=n} x^{(n+1)-i}y^{i-1}
                    \right)\\
                &= (x-y)\left(
                        y^{n}
                        + \sum_{i=1}^{i=n} x^{(n+1)-i}y^{i-1}
                    \right)\\
                &= (x-y)\left(
                        y^{n}
                        + x\sum_{i=1}^{i=n} x^{n-i}y^{i-1}
                    \right)\\
                &= (x-y)\left(
                        y^{n}
                        + xf(x,y,n)
                    \right)\\
\end{split}$$
We can also rewrite what Spivak wants into:
$$
x^n-y^n = (x-y)\Big(y^{n-1} + xf(x,y,n-1)\Big)
$$
We've already proven in [spivak_calc_probs.1.1.2]:
$$\begin{split}
x^2-y^2 &= (x-y)(x+y)\\
        &= (x-y)\left(\sum_{i=1}^{i=2} x^{2-i}y^{i-1}\right)\\
        &= (x-y)\Big(y^{2-1} + xf(x,y,2-1)\Big)\\
\end{split}$$
Then, by induction, we prove that:
$$\begin{split}
x^2-y^2 &= (x-y)\Big(y^{2-1} + xf(x,y,2-1)\Big)\\
x^3-y^3 &= (x-y)\Big(y^{3-1} + xf(x,y,3-1)\Big)\\
\vdots\\
x^n-y^n &= (x-y)\Big(y^{n-1} + xf(x,y,n-1)\Big)\\
\end{split}$$
And since $(x-y)(y^{n-1} + xf(x,y,n-1))$ is only a rewrite of what Spivak
wants, i.e. $(x-y)(x^{n-1}$ $+$ $x^{n-2}y$ $+$ $\ldots$ $+$ $xy^{n-2})$,
therefore Q.E.D already.

alternative proof:
Using the distributive axiom:
$$\begin{split}
& (x-y)\left(\sum_{i=1}^{i=n} x^{n-i}y^{i-1}\right)\\
&= x\left(\sum_{i=1}^{i=n} x^{n-i}y^{i-1}\right)
   -y\left(\sum_{i=1}^{i=n} x^{n-i}y^{i-1}\right)\\
&= x\left(\sum_{i=1}^{i=n} x^{n-i}y^{i-1}\right)
   -y\left(x^{n-n}y^{n-1} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i-1}\right)\\
&= x\left(\sum_{i=1}^{i=n} x^{n-i}y^{i-1}\right)
   -y\left(x^{0}y^{n-1} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i-1}\right)\\
&= x\left(\sum_{i=1}^{i=n} x^{n-i}y^{i-1}\right)
   -y\left(y^{n-1} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i-1}\right)\\
&= x\left(\sum_{i=1}^{i=n} x^{n-i}y^{i-1}\right)
   -\left(y^{n} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i}\right)\\
&= x\left(x^{n-1}y^{1-1} + \sum_{i=2}^{i=n} x^{n-i}y^{i-1}\right)
   -\left(y^{n} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i}\right)\\
&= x\left(x^{n-1}y^{0} + \sum_{i=2}^{i=n} x^{n-i}y^{i-1}\right)
   -\left(y^{n} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i}\right)\\
&= x\left(x^{n-1} + \sum_{i=2}^{i=n} x^{n-i}y^{i-1}\right)
   -\left(y^{n} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i}\right)\\
&= \left(x^{n} + \sum_{i=2}^{i=n} x^{n-i+1}y^{i-1}\right)
   -\left(y^{n} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i}\right)\\
&= \left(x^{n} + \sum_{i=1}^{i=n-1} x^{n-(i+1)+1}y^{(i+1)-1}\right)
   -\left(y^{n} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i}\right)\\
&= \left(x^{n} + \sum_{i=1}^{i=n-1} x^{n-i-1+1}y^{i+1-1}\right)
   -\left(y^{n} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i}\right)\\
&= \left(x^{n} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i}\right)
   -\left(y^{n} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i}\right)\\
&= x^{n} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i}
   - y^{n} - \sum_{i=1}^{i=n-1} x^{n-i}y^{i}\\
\end{split}$$
Then using additive associative axiom:
$$\begin{split}
& x^{n} + \sum_{i=1}^{i=n-1} x^{n-i}y^{i}
   - y^{n} - \sum_{i=1}^{i=n-1} x^{n-i}y^{i}\\
&= x^{n}
   - y^{n}
   + \left(\sum_{i=1}^{i=n-1} x^{n-i}y^{i}
   - \sum_{i=1}^{i=n-1} x^{n-i}y^{i}\right)\\
&= x^{n}
   - y^{n}
   + \left(0\right)\\
&= x^{n}
   - y^{n}
\end{split}$$
 [/INST] $(a - 1)(a^{n-1} + a^{n-2} + ... + 1) =$
$a^n + a^{n-1} + ... + a - (a^{n-1} + a^{n-2} + ... + 1) =$
$a^n - 1$ 
Set a = x/y and multiply by y$^n$.
</s>