First slide

Arithmetico-Geometric Progression

Question

The sum to infinity of the series $S = 1 + \frac{2}{3} + \frac{6}{3^{2}} + \frac{10}{3^{3}} + \frac{14}{3^{4}} + \dots$ is

Moderate

A

4
B

6
C

2
D

3

Solution

Let $S = 1 + \frac{2}{3} + \frac{6}{3^{2}} + \frac{10}{3^{3}} + \frac{14}{3^{4}} + \dots$
$\frac{1}{3} S = \frac{1}{3} + \frac{2}{3^{2}} + \frac{6}{3^{3}} + \frac{10}{3^{4}} + \dots$
Subtracting, we get
$\begin{array}{l} \frac{2}{3} S = 1 + \frac{1}{3} + \frac{4}{3^{2}} + \frac{4}{3^{3}} + \frac{4}{3^{4}} + \dots \\ = 1 + \frac{1}{3} + \frac{4 / 3^{2}}{1 - 1 / 3} = \frac{4}{3} + \frac{2}{3} = 2 \end{array}$
$\Rightarrow S = 3$

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