Calculate Geometric Series from three things

Question

How can I calculate the gp series from third term,third last term and sum of the series?Is the approach of calculating first term by taking gcd of third term and sum of series correct or not?

Answer 1

Let $\{a,ar,ar^2,\ldots,ar^{n-3},ar^{n-2},ar^{n-1}\}$ be a geometric sequence (here $n$ is assumed to be at least 6) with common ratio $r$. Let $x=ar^2$ and $y=ar^{n-3}$. Let $s$ denote the sum \begin{align*} s&=a+ar+ar^2+\cdots+ar^{n-1}\\ &=\frac{x}{r^2}+\frac{x}{r}+x+\cdots+y+yr+yr^2 \end{align*} Given $x,y$, and $s$, we want to determine $a$ and $r$.

Using the well-known formula for geometric series, we obtain the sum $x+xr+\cdots+y$ equals $$\frac{x-y}{1-r}+y$$

So\begin{align*} x(\frac{1}{r^2}+\frac{1}{r})+\frac{x-y}{1-r}+y(1+r+r^2)=s \end{align*} , an equation with only $r$ unknown. Once $r$ is solved, $a$ can be determined using $$a=\frac{x}{r^2}$$ The entire geometric sequence can thus be recovered.