Discussion
A geometric series has common ratio \(\frac{1}{3}\). If the sum of first four terms is 200, then the first term is
Answer: E
The first term is 135.
For a geometric series, \(S_{n} = \frac{a(1 - r^{n})}{1 - r}\). With \(r = \frac{1}{3}\) and n = 4:
\[ \begin{aligned} 200 &= a \cdot \frac{1 - \frac{1}{81}}{\frac{2}{3}} = a \cdot \frac{80}{81} \cdot \frac{3}{2} \\ 200 &= \frac{40a}{27} \\ a &= 135 \end{aligned} \]
Check: 135 + 45 + 15 + 5 = 200. Using the infinite-sum formula \(\frac{a}{1 - r}\) would give \(a = \frac{400}{3}\), not an option.
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