1953 AHSME Problems/Problem 25

Problem

In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. then the common ratio is:

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \text{about }\frac{\sqrt{5}}{2} \qquad \textbf{(C)}\ \frac{\sqrt{5}-1}{2}\qquad \textbf{(D)}\ \frac{1-\sqrt{5}}{2}\qquad \textbf{(E)}\ \frac{2}{\sqrt{5}}$

Solution

Given first term $a$ and common ratio $r$, we have $a=a*r+a*r^2$, and. We divide by $a$ in the first equation to get $1=r+r^2$. Rewriting, we have $r^2+r-1=0$. We use the quadratic formula to get $r = \frac{-1+-\sqrt{1^2-4(1)(-1)}}{2(1)}$. Because the terms all have to be positive, we must add the discriminant, getting an answer of $\frac{\sqrt{5}-1}{2}$ $\boxed{C}$.

See Also

1953 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 24
Followed by
Problem 26
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