Question

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

Answer 1

Let us take a G.P. with terms $a_1,a_2,a_3,a_4,....\infty$ and common ratio $r(|r|<1)$.

Also, let us take the sum of all the terms following each term to be $S_1,S_2,S_3,S_4,....$

Now, $S_1=\frac{a_2}{(1-r)}=\frac{ar}{(1-r)}$,

$S_2=\frac{a_3}{(1-r)}=\frac{a{r}^2}{(1-r)}$

$S_3=\frac{a_4}{(1-r)}=\frac{a{r}^3}{(1-r)}$,

$\Rightarrow \frac{a_1}{S_1}=\frac{a}{\frac{ar}{(1-r)}}=\frac{(1-r)}{r}$,

$\frac{a_2}{S_2}=\frac{ar}{\frac{a{r}^2}{(1-r)}}=\frac{(1-r)}{r}$,

$\frac{a_3}{S_3}=\frac{a{r}^2}{\frac{a{r}^3}{(1-r)}}$

It is clearly seen that the ratio of each term to the sum of all the terms following it is constant.