Question

The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.

Answer 1

$a+ar=5$

$a(1+r)=5$ ...... (i)

$a_n=3(a_{n+1}+a_{n+2}+a_{n+3}+.....)$

$a{r}^{n-1}+3(a{r}^n+a{r}^{n+1}+a{r}^{a+2}+....)$

$a{r}^{n-1}=3a{r}^n(1+r+r^2+....\infty )$

$1=3r\left(\frac{1}{1-r}\right)$

$1-r=3r$

$1=4r$

$r=\frac{1}{4}$

$a(1+r)=5$

$a\left(\frac{5}{4}\right)=5$

$a=4$

Thus the G.P. is $4,1,\frac{1}{4,\frac{1}{16},........}$