JEE MAIN - Mathematics (2025 - 23rd January Morning Shift - No. 16)
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
$-120$
$-1200$
$-1080$
$-1020$
Explanation
$$\begin{aligned}
& \mathrm{a}=3 \\
& \mathrm{~S}_4=\frac{1}{5}\left(\mathrm{~S}_8-\mathrm{S}_4\right) \\
& \Rightarrow 5 \mathrm{~S}_4=\mathrm{S}_8-\mathrm{S}_4 \\
& \Rightarrow 6 \mathrm{~S}_4=\mathrm{S}_8 \\
& \Rightarrow 6 \cdot \frac{4}{2}[2 \times 3+(4-1) \times \mathrm{d}] \\
& =\frac{8}{2}[2 \times 3+(8-1) \mathrm{d}] \\
& \Rightarrow 12(6+3 \mathrm{~d})=4(6+7 \mathrm{~d}) \\
& \Rightarrow 18+9 \mathrm{~d}=6+7 \mathrm{~d} \\
& \Rightarrow \mathrm{~d}=-6 \\
& \mathrm{~S}_{20}=\frac{20}{2}[2 \times 3+(20-1)(-6)] \\
& =10[6-114] \\
& =-1080
\end{aligned}$$
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