Sequences & Series
GP — number of terms satisfying sum inequalities
nta_pyq_2023_jan
Grade 11
Question:
The 4th term of GP is 500 and its common ratio is $\dfrac{1}{m}$, $m \in \mathbb{N}$. Let $S_n$ denote the sum of the first n terms of this GP. If $S_6 > S_5 + 1$ and $S_7 < S_6 + \dfrac{1}{2}$, then the number of possible values of m is ______.
Step-by-Step Solution
Key Concept: Use $T_4 = ar^3 = 500$ to express $a$ in terms of $m$. Then $S_n - S_{n-1} = ar^{n-1}$; so the conditions become $ar^5 > 1$ and $ar^6 < \frac{1}{2}$.
$ar^5 > 1 \Rightarrow \frac{500}{m^2} > 1 \Rightarrow m^2 < 500$. $ar^6 < \frac{1}{2} \Rightarrow \frac{500}{m^3} < \frac{1}{2} \Rightarrow m^3 > 1000 \Rightarrow m > 10$. So $10 < m \leq 22$, giving $m = 11, 12, \ldots, 22$: 12 values.
Correct Answer: 12