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8. Sequences and Series
hard
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
A$-1200$
B$-1080$
C$-1020$
D$-120$
(JEE MAIN-2025)
Solution
$ a =3$
$S_4=\frac{1}{5}\left(S_8- S _4\right)$
$\Rightarrow 5 S_4= S _8- S _4$
$\Rightarrow 6 S_4= S _8$
$\Rightarrow 6 \cdot \frac{4}{2}[2 \times 3+(4-1) \times d ]$
$=\frac{8}{2}[2 \times 3+(8-1) d ]$
$\Rightarrow 12(6+3 d)=4(6+7 d)$
$\Rightarrow 18+9 d=6+7 d$
$\Rightarrow d =-6$
$S_{20}=\frac{20}{2}[2 \times 3+(20-1)(-6)]$
$=10[6-114]$
$=-1080$
$S_4=\frac{1}{5}\left(S_8- S _4\right)$
$\Rightarrow 5 S_4= S _8- S _4$
$\Rightarrow 6 S_4= S _8$
$\Rightarrow 6 \cdot \frac{4}{2}[2 \times 3+(4-1) \times d ]$
$=\frac{8}{2}[2 \times 3+(8-1) d ]$
$\Rightarrow 12(6+3 d)=4(6+7 d)$
$\Rightarrow 18+9 d=6+7 d$
$\Rightarrow d =-6$
$S_{20}=\frac{20}{2}[2 \times 3+(20-1)(-6)]$
$=10[6-114]$
$=-1080$
Standard 11
Mathematics
