Sequences & Series
GP and AP — Finding Position
nta_pyq_2024_apr
Grade 11
Question:
Let the first three terms 2, $p$ and $q$, with $q\neq2$, of a G.P. be respectively the 7th, 8th and 13th terms of an A.P. If the 5th term of the G.P. is the $n$th term of the A.P., then $n$ is equal to:
Step-by-Step Solution
Key Concept: GP: $p^2=2q$. AP: $2=a+6d$, $p=a+7d$, $q=a+12d$. From these: $p-2=d$, $q-p=5d=5(p-2)\Rightarrow q=6p-10$. Also $p^2=2q\Rightarrow p^2-12p+20=0\Rightarrow p=10$ (since $q\neq2$).
$p=10$, $q=50$, $d=8$, $a=-46$. 5th GP term $=1250$. $n=163$.
Correct Answer: 1