JEE Main · 2022 · 29 Jun 2022 · Shift 2 · Sequences and Series

The first series is $3,6,9,12,\dots$ with first term $a_1=3$ and common difference $d_1=3$. The 78th term is $3+77\cdot 3=234$.

The second series is $5,9,13,17,\dots$ with first term $a_2=5$ and common difference $d_2=4$. The 59th term is $5+58\cdot 4=237$.

Common terms must satisfy both $3k$ (multiples of 3) and $5+4m=1+4(m+1)$ (form $4n+1$). The LCM of 3 and 4 is 12, but we need terms of the form $3k$ that are also $\equiv 1(\mathrm{mod}4)$.

The common terms form an AP: $9,21,33,\dots$ with first term $a=9$ and common difference $d=12$.

The largest common term $\le \min (234,237)=234$ is found: $9+(n-1)\cdot 12\le 234\Rightarrow n\le 19.75$, so $n=19$ terms.

Sum = $\frac{19}{2}[2\cdot 9+18\cdot 12]=\frac{19}{2}[18+216]=\frac{19\cdot 234}{2}=2223$.