Minus a Value from a Sum

Question

Think I'm being really stupid. I have the following sum:

$\sum\limits_{n=1}^r x^n$

which is close to a geometric series with the limits being $0$ and $r-1$ (ideally I want to be using that form)

I am trying to get to

$\sum\limits_{n=0}^{r-1} x^n$

I have attempted expanding both to find what is missing but I am still stuck.

Would I need to presumably minus something to get to what I am looking for?

Thanks

Lewis

Answer 1

\begin{align} \sum_{n=0}^{r-1} x^n &= 1 + x + x^2 + \ldots + x^{r-1} \\ &= 1 + \big(x + x^2 + \ldots + x^{r-1} + x^r\big) - x^r \\ &= 1 + \left(\sum_{n=1}^r x^n\right) - x^r \end{align}