First slide

Arithmetico-Geometric Progression

Question

The sum of the infinite series $1 + \frac{2}{3} + \frac{7}{3^{2}} + \frac{12}{3^{3}} + \frac{17}{3^{4}} + \frac{22}{3^{5}} + .........$ is equal to

Moderate

A

$\frac{9}{4}$
B

$\frac{11}{4}$
C

$\frac{15}{4}$
D

$\frac{13}{4}$

Solution

$1 + \frac{1}{3} [2 + \frac{7}{3} + \frac{12}{3^{2}} + . . . . . . . . . . . .]$
$1 + \frac{1}{3} [\frac{2}{1 - \frac{1}{3}} + \frac{\frac{5}{3}}{{(1 - \frac{1}{3})}^{2}}] [s u m o f i n f i n i t e t e r m s i n A G P = \frac{a}{1 - r} + \frac{d \cdot r}{{(1 - r)}^{2}}]$
$1 + \frac{1}{3} [3 + \frac{15}{4}] = \frac{13}{4}$

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