JEE Main 2025 — Sequences and Series Question with Solution

$\begin{aligned} & S_{40}=1030 \Rightarrow \frac{40}{2}[2 a+39 d]=1030 \\ & \Rightarrow \quad 2 a+39 d=\frac{103}{2}....(1) \\ & \quad S_{12}=57 \Rightarrow \frac{12}{2}[2 a+11 d]=57 \\ & \Rightarrow \quad 2 a+11 d=\frac{57}{6}...(2) \end{aligned}$ Equation (1) - equation (2) $\begin{aligned} & 28 d=\frac{103}{2}-\frac{57}{6} \\ & 28 d=\frac{309-57}{6} \end{aligned}$ $d=\frac{3}{2}$ $\Rightarrow a=-\frac{7}{2}$ $S_{30}-S_{10}=\frac{30}{2}[2a+29d]-\frac{10}{2}[2a+9d]$ $=15[2a+29d]-5[2a+9d]$ $=5[6a+87d-2a-9d]$ $=5[4a+78d]$ $=5[-14+117]$ $=515$