JEE Main 2025 (Online) 23rd January Morning ShiftQuestion 1
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
Sequence and Series · Arithmetic Progression
Solution
Formulas used. (1) $S_n=\frac n2\left[2a+(n-1)d\right]$(2) Sum of the next four terms $=S_8-S_4$Step 1 - The condition. $$S_4=\frac15\left(S_8-S_4\right)\ \Longrightarrow\ S_8=6S_4 .$$ Step 2 - Substitute $a=3$. $$4(6+7d)=6\cdot2(6+3d)\ \Longrightarrow\ 24+28d=72+36d\ \Longrightarrow\ d=-6 .$$ Step 3 - Sum of $20$ terms. $$S_{20}=10\left[6+19(-6)\right]=10(6-114)=-1080 .$$ Answer: (C)