JEE Main 2024 — Sequences and Series Question with Solution

$4,9,14,19,\dots$, up to ${25}^{\text{th }}$ term

$\Rightarrow {\mathrm{T}}_{25}=4+(25-1)5$

$\Rightarrow {\mathrm{T}}_{25}=4+120$

$\Rightarrow {\mathrm{T}}_{25}=124$

$3,6,9,12,\dots$, up to ${37}^{\text{th }}$ term

$\Rightarrow {\mathrm{T}}_{37}=3+(37-1)3$

$\Rightarrow {\mathrm{T}}_{37}=3+108$

$\Rightarrow {\mathrm{T}}_{37}=111$

Common difference of ${\mathrm{I}}^{\text{st }}$ series is ${\mathrm{d}}_1=5$

Common difference of ${\mathrm{II}}^{\text{nd }}$ series is ${\mathrm{d}}_2=3$

First common term $=9$, and their common difference $=15(\mathrm{LCM}$ of ${\mathrm{d}}_1$ and $\left.{\mathrm{d}}_2\right)$.

So, the common terms are $9, 24, 39, 54, 69, 84, 99$.