If the first term of an A.P. is 3 and the sum of its first four...

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

Options

A $-1080$
B $-1020$
C $-1200$
D $-120$

Correct answer

A. $-1080$

Step-by-step solution

\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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