Discussion
In a geometric progression of positive terms, if any term is equal to the sum of the next two terms, then the common ratio of the geometric progression is equal to
Answer: B
The common ratio is \(\frac{\sqrt{5} - 1}{2}\).
Let the term be \(a\), so the next two are \(ar\) and \(ar^{2}\). Then
\[ \begin{aligned} a &= ar + ar^{2} \\ r^{2} + r - 1 &= 0 \\ r &= \frac{-1 \pm \sqrt{5}}{2} \end{aligned} \]
The terms are positive, so \(r \gt 0\), which leaves \(r = \frac{\sqrt{5} - 1}{2} \approx 0.618\). \(\frac{\sqrt{5} + 1}{2}\) is the reciprocal (the golden ratio), which would make each term the sum of the two before it rather than the two after it.
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