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 $\frac{\sqrt{5}-1}{2}$ $\boxed{C}$ .

1953 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Problem 26
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
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1953 AHSME Problems/Problem 25

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