How do you find the value of the ratio in a series?

Question

The sum of the geometric series $a,ar,ar^2,\dots,ar^{n-1}$ is $$s=\frac{a(r^n-1)}{r-1}$$ Given $a,s,n,r$ we can easily solve for $a,s,n$. But how can we solve for $r$ in terms of $a,s,n$? Also $r$ will always be between -1 and 1 What method should I use to find $r$ and how does the method work?

Answer 1

If $n \leq 5$ there are formulas to solve this, though the one for $n=5$ is a mess. For larger $n$ you are in for a numeric solution. I find fixed point iteration handy. For many values of $a,s,n$ you could write $$r_{i+1}=\sqrt[n]{\frac sa(r_i-1)+1}$$ because roots do not change too much. If $n$ is large and $r \gt 1$ the last term will dominate, so I would ignore all the early terms in the sum and try $r_0=\sqrt[n-1]{\frac sa}$