In series A, the first term is 2, and the common ratio is 0.5. In series B, the first term is 3, and the two infinite series have the same sum. What is the ratio in series B?

Question

Answer 1

Given
First term of series $A - a = 2$
Common ratio of series $A - r_{A} = 0.5$
First term of series $B - b = 3$ Let common ratio of series $B - r_{b}$
Sum of infinite geometric series is given by-
$S_{\infty} = \frac{a}{1 - r}$
Sum of series A-
$S_{A \infty} = \frac{a}{1 - r_{a}}$
Sum of series B-
$S_{B \infty} = \frac{b}{1 - r_{b}}$
Sum of both the series are same.
$∴ S_{A \infty} = S_{B \infty}$
$\Rightarrow \frac{a}{1 - r_{a}} = \frac{b}{1 - r_{b}}$
$\Rightarrow \frac{2}{1 - 0.5} = \frac{3}{1 - r_{b}}$
$\Rightarrow 4 = \frac{3}{1 - r_{b}}$
$\Rightarrow 1 - r_{b} = \frac{3}{4}$
$\Rightarrow r_{b} = \frac{1}{4}$
$\Rightarrow r_{b} = 0.25$
Common ratio of series B is 0.25