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]

Question

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]

Smart Achievers · Accepted Answer

(12)

Let the first term be $A$

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

$S_{6} > S_{5} + 1$

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

$\Rightarrow m \leq 22$ $(∵ m \in ℕ)$

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

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

$Thus, 11 \leq m \leq 22$

$∴$ $m can take 12 values.$

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]

Want Us to Email you About Special Offers & Updates?

Solution

Q. The 4th term of GP is 500 and its common ratio is 1/m,m∈N. Let Sn denote the sum of the first n terms of this GP. If S6>S5+1 and S7<S6+12, then the number of possible values of m is _______ . [2023]

Ans. (12) Let the first term be A T4=500 ; Am3=500 S6-S5>1⇒T6>1⇒Am5>1 ⇒500m2>1⇒m2<500⇒m<500 ⇒m≤22 (∵m∈ℕ) S7-S6<12⇒T7<12⇒ Am6<12⇒500m3<12 ⇒m3>1000⇒m≥11 Thus, 11≤m≤22 ∴ m can take 12 values.

Want Us to Email you About Special Offers & Updates?

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]