Let the first three terms 2, p and q , with q neq 2 , of a G.P. be respectively the 7 th , 8 th and 13 th terms of an A.P. If the 5 th term of the G.P. is the n th term of the A.P., then n is equal to

Mathematics · Sequence and Series

JEE Main 2024 — 4 April, Shift 1 — Question 12

Let the first three terms $2, p$ and $q$ , with $q \neq 2$ , of a G.P. be respectively the $7^{\text {th }}, 8^{\text {th }}$ and $13^{\text {th }}$ terms of an A.P. If the $5^{\text {th }}$ term of the G.P. is the $\mathrm{n}^{\text {th }}$ term of the A.P., then $n$ is equal to

Option A:
151
Option B:
169
Option C:
177
Option D:
163
Correct

Answer: D

Step-by-step solution

$\mathrm{p}^{2}=2 \mathrm{q}$

$2=a+6 d \quad \ldots(i)$

$\mathrm{p}=\mathrm{a}+7 \mathrm{~d} \quad \ldots(ii)$

$\mathrm{q}=\mathrm{a}+12 \mathrm{~d} \ldots (iii)$

$\mathrm{p}-2=\mathrm{d} \quad($ (ii) - (i)) $\mathrm{q}-\mathrm{p}=5 \mathrm{~d})$

$\mathrm{q}-\mathrm{p}=5(\mathrm{p}-2)$

$\mathrm{q}=6 \mathrm{p}-10$

$\mathrm{p}^{2}=2(6 \mathrm{p}-10)$

$\mathrm{p}^{2}-12 \mathrm{p}+20=0$

$\mathrm{p}=10,2$ $\mathrm{p}=10 ; \mathrm{q}=50$

$\mathrm{d}=8$

$a=-46$

$2,10,50,250,1250$

$\mathrm{ar}^{4}=\mathrm{a}+(\mathrm{n}-1) \mathrm{d}$

$1250=-46+(\mathrm{n}-1) 8$

$\mathrm{n}=163$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 4 April, Shift 1
Subject: Mathematics
Chapter: Sequence and Series
Topic: Geometric Progression