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.

\(\therefore S_{A\infty}=S_{B\infty}\)

\(\Rightarrow\frac{a}{1-r_a}=\frac{b}{1-r_b}\)

\(\Rightarrow\frac{2}{1-0.5}=\frac{3}{1-r_b}\)

\(\Rightarrow4=\frac{3}{1-r_b}\)

\(\Rightarrow1-r_b=\frac{3}{4}\)

\(\Rightarrow r_b=\frac{1}{4}\)

\(\Rightarrow r_b=0.25\)

Common ratio of series B is 0.25

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.

\(\therefore S_{A\infty}=S_{B\infty}\)

\(\Rightarrow\frac{a}{1-r_a}=\frac{b}{1-r_b}\)

\(\Rightarrow\frac{2}{1-0.5}=\frac{3}{1-r_b}\)

\(\Rightarrow4=\frac{3}{1-r_b}\)

\(\Rightarrow1-r_b=\frac{3}{4}\)

\(\Rightarrow r_b=\frac{1}{4}\)

\(\Rightarrow r_b=0.25\)

Common ratio of series B is 0.25