Discussion
The sum of the first \(n\) terms in a G.P. is \(s_{n} = 100 - 100^{1-n}\) . The common ratio r is
Answer: E
The common ratio is \(\dfrac{1}{100}\).
Rewrite the sum in the standard form \(S_{n} = \frac{a(1 - r^{n})}{1 - r}\):
\[ \begin{aligned} s_{n} &= 100 - 100 \cdot 100^{-n} \\ &= 100\left(1 - \left(\tfrac{1}{100}\right)^{n}\right) \end{aligned} \]
The term raised to the power \(n\) is the common ratio, so \(r = \frac{1}{100}\). Check: \(a = s_{1} = 100 - 1 = 99\), and \(\frac{a}{1 - r} = \frac{99}{0.99} = 100\), matching the factor in front. The other fractions do not reproduce \(s_{1} = 99\).
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