JEE PYQ: Sequence And Series Question 15
Question 15 - 2021 (26 Feb Shift 1)
The sum of the infinite series $1 + \frac{2}{3} + \frac{7}{3^2} + \frac{12}{3^3} + \frac{17}{3^4} + \frac{22}{3^5} + \ldots$ is equal to
(1) $\frac{9}{4}$
(2) $\frac{15}{4}$
(3) $\frac{13}{4}$
(4) $\frac{11}{4}$
Show Answer
Answer: (3)
Solution
$s = 1 + \frac{2}{3} + \frac{7}{3^2} + \frac{12}{3^3} + \frac{17}{3^4} + \ldots$
$\frac{s}{3} = \frac{1}{3} + \frac{2}{3^2} + \frac{7}{3^3} + \ldots$
$\frac{2s}{3} = 1 + \frac{1}{3} + \frac{5}{3^2} + \frac{5}{3^3} + \ldots \infty$
$\frac{2s}{3} = \frac{4}{3} + \frac{5}{3}\left{\frac{1/3}{1-\frac{1}{3}}\right} = \frac{5}{6} + \frac{4}{3} = \frac{13}{6}$
$s = \frac{13}{4}$
BETA
