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.