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8. Sequences and Series
hard
The $4^{\text {tht }}$ term of $GP$ is $500$ and its common ratio is $\frac{1}{m}, m \in N$. Let $S_n$ denote the sum of the first $n$ terms of this GP. If $S_6 > S_5+1$ and $S_7 < S_6+\frac{1}{2}$, then the number of possible values of $m$ is $..........$
A
$11$
B
$10$
C
$12$
D
$15$
(JEE MAIN-2023)
Solution
$T_4=500 \quad$ where $a=$ first term,
$r =$ common ratio $=\frac{1}{ m }, m \in N$
$a r^3=500$
$\frac{a}{m^3}=500$
$S_n-S_{n-1}=a r^{n-1}$
$S _6 > S _5+1 \quad$ and $S _7- S _6 < \frac{1}{2}$
$S _6- S _5 > 1 \quad \frac{ a }{ m ^6} < \frac{1}{2}$
$ar ^5 > 1 \quad m ^3 > 10^3$
$\frac{500}{ m ^2} > 1 \quad m > 10$
$m ^2 < 500$
From $(1)$ and $(2)$
$m =11,12,13 \ldots \ldots \ldots \ldots ., 22$
So number of possible values of $m$ is $12$
Standard 11
Mathematics
