1953 AMC 12 第 25 题

Question

一个各项均为正数的等比数列中，每一项都等于它后面两项之和。则公比为：

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:

1

约为

\dfrac{\sqrt5}{2}

about

\dfrac{\sqrt5}{2}

\dfrac{\sqrt5-1}{2}

\dfrac{1-\sqrt5}{2}

\dfrac2{\sqrt5}

将三个连续项之间的等式除以其中第一项

Divide the relation among three consecutive terms by the first of them

公比满足

1=r+r^2

；取正根

The common ratio satisfies

1=r+r^2

; choose its positive root

若某一项为

t

，则后面两项为

tr

和

tr^2

。因此

t=tr+tr^2

且

r^2+r-1=0

。因为各项均为正数，所以

r=\frac{\sqrt5-1}{2}\text{。}

因此，正确答案是 C。

If a term is

t,

the next two are

tr

and

tr^2.

Thus

t=tr+tr^2

and

r^2+r-1=0.

Because the terms are positive,

r=\frac{\sqrt5-1}{2}.

Thus, the correct answer is C.

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