# 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
Series
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?​

2021-03-08
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

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