Question

The $4^{\text {th }}$ term of $G P$ 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

Answer 1

The correct answer is 12.
T4​=500 where a= first term,
r= common ratio =m1​,m∈N
ar3=500
m3a​=500
Sn​−Sn−1​=arn−1
S6​>S5​+1
and S7​−S6​<21​
S6​−S5​>1m6a​<21​
ar5>1m3>103
m2500​>1m>10...(2)
m2<500.........(1)
From (1) and (2)
m=11,12,13………….,22
So number of possible values of m is 12