Finding a finite sum of an alternating geometric series

Question

I'm trying to find the summation of a finite alternating geometric series. I know that normally the sum of a geometric series can be calculated using $a\frac{1-r^n}{1-r}$. This same equation doesn't work for an alternating geometric series such as $(-2^n)$, where the series is $1,-2,4,-8,16$. I'm looking to find the summation of the first 50 terms.

Answer 1

The same formula works. Evaluate $$\frac{a(1-r^n)}{1-r}$$ at $r=-2$.

For example when $n=3$, it is $$\frac{(1)(1-(-2)^3)}{1-(-2)}=3=1-2+4$$

Answer 2

You could use the fact that the sum of two consecutive terms is $-2^{n-1}$, where you are adding the nth term and the n+1st term, so the sum of the first 50 terms is $\sum_{n=1}^{25}-2^{n-1}$, and go from there