Question

Show that in an infinite $G.P.$ with common ratios $(|r|<1)$ , each term bears a constant ratio to the sum of all terms that follow it.

Answer 1

Let the G.P be $a,ar,a{r}^2,a{r}^3$,.....

Consider a general term $a.r^{n-1}$

Sum of all the terms after equals

$a.r^n+a.r^{n+1}+a.r^{n+2}+$.....

$=\frac{a.r^n}{1-r}$

Ratio of the term and the sum of following terms $=\frac{a.r^{n-1}}{\frac{a.r^n}{1-r}}$

$=\frac{(1-r).a.r^{n-1}}{a.r^n}$

$=\frac{1-r}{r}$

which does not depend on 'n'.

Hence the proof.