Discussion
A geometric progression has an even number of terms. If the sum of all terms is five times the sum of all odd terms, then the common ratio is equal to
Answer: A
The common ratio is 4.
With an even number of terms, the terms pair up as (1st, 2nd), (3rd, 4th), and so on. Each even-placed term is \(r\) times the odd-placed term before it, so
\[ \begin{aligned} S_{\text{even}} &= r\,S_{\text{odd}} \\ S_{\text{all}} &= S_{\text{odd}} + r\,S_{\text{odd}} = (1 + r)S_{\text{odd}} \end{aligned} \]
Given \(S_{\text{all}} = 5S_{\text{odd}}\), \(1 + r = 5\) and \(r = 4\).
\(\frac{1}{5}\) and \(\frac{4}{5}\) invert the ratio of the sums, and 2 would make the total three times the odd-term sum.
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