JEE PYQ: Sequence And Series Question 44
Question 44 - 2020 (09 Jan Shift 2)
Let $a_n$ be the $n^{th}$ term of a G.P. of positive terms. If $\sum_{n=1}^{100} a_{2n+1} = 200$ and $\sum_{n=1}^{100} a_{2n} = 100$, then $\sum_{n=1}^{200} a_n$ is equal to:
(1) 300
(2) 225
(3) 175
(4) 150
Show Answer
Answer: (4)
Solution
Let G.P. be $a, ar, ar^2, \ldots$
$\sum_{n=1}^{100} a_{2n+1} = a_3 + a_5 + \ldots + a_{201} = 200$
$\sum_{n=1}^{100} a_{2n} = a_2 + a_4 + \ldots + a_{200} = 100$
From equations (i) and (ii), $r = 2$ and
$a_2 + a_3 + \ldots + a_{200} + a_{201} = 300$
$\Rightarrow r(a_1 + \ldots + a_{200}) = 300$
$\Rightarrow \sum_{n=1}^{200} a_n = \frac{300}{r} = 150$
BETA
