Calculate

$$
P=1+\pi^{1}+\pi^{2}+\pi^{3}+\ldots+\pi^{10}
$$

using

$$
\pi \times P=\pi^{1}+\pi^{2}+\ldots+\pi^{11}
$$
$$
\begin{gathered}
\pi \times P-P=\pi^{11}-1 \\
P=\frac{\pi^{11}-1}{\pi-1}
\end{gathered}
$$

As you can see, we can replace $\pi$ with any other integer, real, or even complex number, PROVIDED that we do not divide by zero. That is, we cannot calculate

$$
1^{0}+1^{1}+1^{2}+1^{3}+\ldots+1^{n}
$$

using this technique. However, this is not very difficult to calculate.

From these statements, we deduce a non-trivial factorization:

Proposition 6.

$$
\left(x^{n}-1\right)=(x-1)\left(1+x+x^{2}+x^{3}+\ldots+x^{n-1}\right)
$$

and even by setting $x=a / b$

$$
a^{n}-b^{n}=(a-b)\left(a^{n-1}+a^{n-2} b+\ldots+a b^{n-2}+b^{n-1}\right)
$$