Question

Using geometric series, why is the polynomial `1+x+x^2+x^3` be written as `(x^4-1)/(x-1)` assuming x does not equal 1?

Answer 1

This is a series with first term `1`, common ratio `x` and `4` terms.

The formula for sum of a geometric series is `s_n = (a(1 - r^n))/(1 - r)`.

`s_4= (1(1 - x^4))/(1 - x)`

`s_4 = (1 - x^4)/(1 - x)`

`s_4 = (x^4 - 1)/(x- 1)`

Proved!!

Hopefully this helps!