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:
Step-by-Step Solution
Key Concept: `Sum of next four' $=S_{8}-S_{4}$. The relation $S_{4}=\tfrac{1}{5}(S_{8}-S_{4})$ becomes $6S_{4}=S_{8}$, a single linear equation in $d$.
Given $a=3$ and $S_{4}=\dfrac{1}{5}(S_{8}-S_{4})$, i.e.\ $6S_{4}=S_{8}$.
$$6\cdot\frac{4}{2}\bigl[2\cdot 3+3d\bigr]=\frac{8}{2}\bigl[2\cdot 3+7d\bigr]\ \Longrightarrow\ 12(6+3d)=4(6+7d).$$
$$72+36d=24+28d\ \Longrightarrow\ 8d=-48\ \Longrightarrow\ d=-6.$$
Hence
$$S_{20}=\frac{20}{2}\bigl[2\cdot 3+19(-6)\bigr]=10[6-114]=-1080.$$
Correct Answer: 1