Question

The sum of an infinite geometric series is $3$. When the common ratio of the series is doubled, then the sum becomes $5$. The first term of the series is

• A. $\frac{8}{7}$
• B. $\frac{15}{7}$
• C. $\frac{17}{7}$
• D. $\frac{2}{7}$

Answer 1

The correct option is B $\frac{15}{7}$

Let the first term be $a$ and the common ratio be $r$,
$\frac{a}{1-r}=\mathrm{3.....}\left(i\right)$
$\frac{a}{1-2r}=\mathrm{5.....}\left(ii\right)$
Dividing the two equation,
$\frac{1-2r}{1-r}=\frac{3}{5}\Rightarrow 5-10r=3-3r\Rightarrow r=\frac{2}{7}$
Using equation $\left(i\right)$,
$\frac{a}{1-\frac{2}{7}}=3\Rightarrow a=\frac{15}{7}$