The 4^{text {tht }} term of GP is 500 and its | vedclass

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Question

$T_4=500 \quad$ 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}

Vedclass · Accepted Answer

$12$

Vedclass · Answer

$T_4=500 \quad$ 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 $12$

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 $..........$

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