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JEE Main 20244 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
4 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Sequences and Series

JEE Main 2024 · 4 April · Morning Shift · Mathematics · Sequences and Series

Let the first three terms 2, p and q , with q ≠ 2 , of a G.P. be...

Let the first three terms 2,p2, p and qq, with q≠2q \neq 2, of a G.P. be respectively the 7th ,8th 7^{\text{th }}, 8^{\text{th }} and 13th 13^{\text{th }} terms of an A.P. If the 5th 5^{\text{th }} term of the G.P. is the nth n^{\text{th }} term of the A.P., then nn is equal to:

Options

  1. A163
  2. B151
  3. C177
  4. D169

Correct answer

A. 163

Step-by-step solution

p2=2q2=a+6dp^2=2 q 2=a+6 d ...(i) p=a+7 d\mathrm{p}=\mathrm{a}+7 \mathrm{~d} ...(ii) q=a+12dq=a+12 d ...(iii) p−2=d\mathrm{p}-2=\mathrm{d} ((ii) - (i)) q−p=5dq-p=5 d ((iii) - (ii)) q−p=5dq-p=5 dq−p=5(p−2)q=6p−10p2=2(6p−10)p2−12p+20=0p=10,2p=10;q=50d=8a=−462,10,50,250,1250a4=a+(n−1)d1250=−46+(n−1)8n=163\begin{aligned} & q-p=5(p-2) \\ & q=6 p-10 \\ & p^2=2(6 p-10) \\ & p^2-12 p+20=0 \\ & p=10,2 \\ & p=10 ; q=50 \\ & d=8 \\ & a=-46 \\ & 2,10,50,250,1250 \\ & a^4=a+(n-1) d \\ & 1250=-46+(n-1) 8 \\ & n=163\end{aligned}$$
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