JEE Main 2025 — Sequences and Series Question with Solution
JEE Main 2025 (29 Jan Shift 1)
Question
Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its term is :
Choose an option
Show full solutionCorrect option: A
Correct answer
A90
Step-by-step explanation
$\begin{aligned}
& \mathrm{S}_3=3 \mathrm{a}+3 \mathrm{~d}=54 \\
& \Rightarrow \mathrm{a}+\mathrm{d}=18 \\
& \mathrm{~S}_{20}=10(2 \mathrm{a}+19 \mathrm{~d}) \\
& \Rightarrow 10(36+17 \mathrm{~d}) \\
& \Rightarrow 1600 < 10(36+17 \mathrm{~d}) < 1800 \\
& \Rightarrow 160 < 36+17 \mathrm{~d} < 180 \\
& \Rightarrow 124 < 17 \mathrm{~d} < 144 \\
& \Rightarrow 7 \frac{5}{17} < \mathrm{d} < 8 \frac{8}{17}
\end{aligned}$
Common difference will be natural number
$\begin{aligned}
& \Rightarrow d=8 \Rightarrow a=10 \\
& \Rightarrow a_{11}=10+10 \times 8=90
\end{aligned}$
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This is a previous-year question from JEE Main 2025, covering the Sequences and Series chapter of Mathematics. PrepSharp catalogues every PYQ from JEE Main with a verified answer key and step-by-step solution prepared by IIT alumni — so you can search by chapter, topic or year and revise efficiently.