Mathematics · Sequence and Series

JEE Main 2025 — 23 January, Morning Shift — Question 8

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

Option A:
$-1200$
Option B:
$-1080$
Correct
Option C:
$-1020$
Option D:
$-120$

Answer: B

Step-by-step solution

$\mathrm{a}=3$

$\mathrm{S}_{4}=\frac{1}{5}\left(\mathrm{~S}_{8}-\mathrm{S}_{4}\right)$

$\Rightarrow 5 \mathrm{~S}_{4}=\mathrm{S}_{8}-\mathrm{S}_{4}$

$\Rightarrow 6 \mathrm{~S}_{4}=\mathrm{S}_{8}$ $\Rightarrow 6 . \frac{4}{2}[2 \times 3+(4-1) \times d]$

$=\frac{8}{2}[2 \times 3+(8-1) \mathrm{d}]$

$\Rightarrow 12(6+3 \mathrm{~d})=4(6+7 \mathrm{~d})$

$\Rightarrow 18+9 \mathrm{~d}=6+7 \mathrm{~d}$

$\Rightarrow \mathrm{d}=-6$

$\mathrm{S}_{20}=\frac{20}{2}[2 \times 3+(20-1)(-6)]$

$=10[6-114]$

$=-1080$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 23 January, Morning Shift
Subject: Mathematics
Chapter: Sequence and Series
Topic: Arithmetic Progression