1976 AMC 12 第 4 题

Question

一个有

n

项的等比数列首项为一、公比为

r

、和为

s

，其中

r

和

s

均不为零。把原数列的每一项替换为其倒数后所得等比数列的和为

Let a geometric progression with

n

terms have first term one, common ratio

r

and sum

s,

where

r

and

s

are not zero. The sum of the geometric progression formed by replacing each term of the original progression by its reciprocal is

\frac1s

\frac1{r^ns}

\frac{s}{r^{n-1}}

\frac{r^n}{s}

\frac{r^{n-1}}s

把倒数数列的和写成

1+r^{-1}+\cdots+r^{-(n-1)}

Write the reciprocal sum as

1+r^{-1}+\cdots+r^{-(n-1)}

将幂的顺序倒过来后，提取因子

r^{-(n-1)}

Factor

r^{-(n-1)}

after reversing the order of the powers

设倒数数列的和为

S'

。乘以

r^{n-1}

后，原数列各项的顺序恰好反转：

\begin{aligned} r^{n-1}S' &=r^{n-1}+r^{n-2}+\cdots+1\\ &=s \end{aligned}\text{。}

因此

S'=\frac{s}{r^{n-1}}

。当

r=1

时结论也成立。

所以正确答案是 C。

Let

S'

be the reciprocal sum. Multiplying it by

r^{n-1}

reverses the original terms:

\begin{aligned} r^{n-1}S' &=r^{n-1}+r^{n-2}+\cdots+1\\ &=s. \end{aligned}

Hence

S'=\frac{s}{r^{n-1}}.

This also holds when

r=1.

Therefore, the correct answer is C.

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