Question

The first term of an infinite geometric series is $6$ and its sum is $8$. Find the G.P.

Answer 1

We know that the formula for sum of infinite terms of G.P is $S_{\infty }=\frac{a}{1-r}$, where $a$ is the first term, $r$ is the common ratio.

It is given that the first term is $a=6$ and the sum of the infinite geometric series is $S_{\infty }=8$, therefore,

$S_{\infty }=\frac{a}{1-r}\Rightarrow 8=\frac{6}{1-r}\Rightarrow 1-r=\frac{6}{8}=\frac{3}{4}\Rightarrow r=1-\frac{3}{4}\Rightarrow r=\frac{4-3}{4}=\frac{1}{4}$

Therefore, with $a=6$ and $r=\frac{1}{4}$, the terms of G.P are:

$a_1=6a_2=a_1\times r=6\times \frac{1}{4}=\frac{3}{2}{a}_3=a_2\times r=\frac{3}{2}\times \frac{1}{4}=\frac{3}{8}{a}_4=a_3\times r=\frac{3}{8}\times \frac{1}{4}=\frac{3}{32}$

Hence, the G.P is $6,\frac{3}{2},\frac{3}{8},\frac{3}{32},.....$.