Sequences & Series
Two GPs — Equal Terms
nta_pyq_2024_jan
Grade 11
Question:
Let $a$ and $b$ be two distinct positive real numbers. Let $11^{\text{th}}$ term of a GP, whose first term is $a$ and third term is $b$, be equal to $p^{\text{th}}$ term of another GP, whose first term is $a$ and fifth term is $b$. Then $p$ is equal to
Step-by-Step Solution
Key Concept: GP1: first $a$, 3rd $b\Rightarrow r_1^2=b/a$. $t_{11}=ar_1^{10}=a(b/a)^5$. GP2: first $a$, 5th $b\Rightarrow r_2^4=b/a$. $T_p=ar_2^{p-1}=a(b/a)^{(p-1)/4}$. Set equal: $(b/a)^5=(b/a)^{(p-1)/4}\Rightarrow(p-1)/4=5\Rightarrow p=21$.
$p=21$.
Correct Answer: 3