The 4 t h term of GP is 500 and its common ratio is 1 / m , m 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 + 1 2 , then the number of possible values of m is _______ . [2023]

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Solution

Q.
The $4^{t h}$ term of GP is 500 and its common ratio is $1 / m, m \in N .$ 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 _______ . [2023]

Ans.
(12)

Let the first term be $A$

$T_{4} = 500; \frac{A}{m^{3}} = 500$

$S_{6} - S_{5} > 1 \Rightarrow T_{6} > 1 \Rightarrow \frac{A}{m^{5}} > 1$

$\Rightarrow \frac{500}{m^{2}} > 1 \Rightarrow m^{2} < 500 \Rightarrow m < \sqrt{500}$

$\Rightarrow m \leq 22$ $(∵ m \in ℕ)$

$S_{7} - S_{6} < \frac{1}{2} \Rightarrow T_{7} < \frac{1}{2} \Rightarrow \frac{A}{m^{6}} < \frac{1}{2} \Rightarrow \frac{500}{m^{3}} < \frac{1}{2}$

$\Rightarrow m^{3} > 1000 \Rightarrow m \geq 11$

$Thus, 11 \leq m \leq 22$

$∴$ $m can take 12 values.$

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