Question

(iv) If the sum to infinity of a G. P. is 9 and sum of first two terms is 5, then find the common ratio. ​

Answer 1

Given,

Sum of infinity terms of G.P. $\[ = 9\]$

Sum of first two terms of G.P. $\[ = 5\]$

Solution,

Consider the first term of G.P. is a and common ratio is r.

According to first condition,

$\[\frac{a}{{1 - r}} = 9\]$------(1)

According to second condition,

$\[a + ar = 5\]$------(2)

Put the value of a from equation 1 into equation 2,

$\[a\left( {1 + r} \right) = 5\]$

$\[ \Rightarrow 9\left( {1 - r} \right)\left( {1 + r} \right) = 5\]$

$\[ \Rightarrow \left( {1 - {r^2}} \right) = \frac{5}{9}\]$

Simplify it,

$\[ \Rightarrow {r^2} = 1 - \frac{5}{9}\]$

$\[ \Rightarrow {r^2} = \frac{4}{9}\]$

$\[ \Rightarrow r = \pm \frac{2}{3}\]$

Hence the common ratio of G.P. is $\[\left( {\frac{2}{3}} \right)\]$.