Finding the first term of a geometric series by the sum and $n$

Question

I have the following exponential series:

$$S = ar^0 + ar^1 + ar^2 + \cdots + ar^n$$

I know $S$, $r$ and $n$. How do I find $a$?

I actually need this done by a script so all "crazy" methods like doing an operation $n$ times are ok.

user9413 · Accepted Answer · 2011-04-14 12:34:45Z

up vote 7 down vote accepted

As Raskolnikov points out this is a geometric series. Sum is given by $$S = a \cdot \frac{r^{n+1}-1}{r-1}$$

Substitute the value of $S,r,n$ to get $a$

answered Apr 14 '11 at 12:34

user9413

	actually by the link Raskolnikov gave its $S = a \cdot \frac{r^{n+1}-1}{r-1}$ because my series terminates in $n$ not $n-1$ – Dani Apr 14 '11 at 12:36
	@Dani: Yes, Dani because here there are $n+1$ terms. The formula which I gave was for $n$ terms. – user9413 Apr 14 '11 at 12:38

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