JEE PYQ: Sequence And Series Question 3
Question 3 - 2021 (16 Mar Shift 2)
Let $S_n(x) = \log_{a^{1/2}} x + \log_{a^{1/3}} x + \log_{a^{1/6}} x + \log_{a^{1/11}} x + \log_{a^{1/18}} x + \log_{a^{1/27}} x + \ldots$ up to $n$ terms, where $a > 1$. If $S_{24}(x) = 1093$ and $S_{12}(2x) = 265$, then value of $a$ is equal to _______.
Show Answer
Answer: (16)
Solution
$S_n(x) = (2 + 3 + 6 + 11 + 18 + 27 + \ldots + n \text{ terms})\log_a x$
Let $S_1 = 2 + 3 + 6 + 11 + 18 + 27 + \ldots + T_n$
$T_n = 2 + (n-1)^2$
$S_1 = \Sigma T_n = 2n + \frac{(n-1)n(2n-1)}{6}$
$\Rightarrow S_n(x) = \left(2n + \frac{n(n-1)(2n-1)}{6}\right)\log_a x$
$S_{24}(x) = 1093$ (Given)
$\log_a x = \frac{1}{4}$ …(1)
$S_{12}(2x) = 265$
$\log_a 2x = \frac{1}{2}$ …(2)
$(2) - (1)$:
$\log_a 2x - \log_a x = \frac{1}{4}$
$\log_a 2 = \frac{1}{4} \Rightarrow a = 16$
BETA
