1959 AMC 12 第 12 题

Question

分别给

20

、

50

、

100

加上同一个常数后，所得三数构成等比数列。其公比为：

By adding the same constant to each of

20,

50,

100

a geometric progression results. The common ratio is:

\dfrac53

\dfrac43

\dfrac32

\dfrac12

\dfrac13

设所加常数为

c

Let

c

be the added constant

对三个连续的等比数列项，中间项的平方等于两端项之积

For three consecutive geometric terms, the square of the middle term equals the product of the outer terms

等比数列的条件为

(50+c)^2=(20+c)(100+c)\text{。}

展开并消去

c^2

得

500=20c

，所以

c=25

。公比为

\frac{50+25}{20+25}=\frac{75}{45}=\frac53\text{。}

因此，正确答案是 A。

The geometric-progression condition is

(50+c)^2=(20+c)(100+c).

Expanding and cancelling

c^2

gives

500=20c,

so

c=25.

The common ratio is

\frac{50+25}{20+25}=\frac{75}{45}=\frac53.

Thus, the correct answer is A.

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