1974 AMC 12 第 21 题

Question

在一个各项均为正数的等比数列中，第五项与第四项之差为

576

，第二项与第一项之差为

9

。这个数列前五项的和是多少？

In a geometric series of positive terms the difference between the fifth and fourth terms is

576,

and the difference between the second and first terms is

9.

What is the sum of the first five terms of this series?

1061

1023

1024

768

以上皆非

none of these

设首项为

a

，公比为

r

Write the first term as

a

and the common ratio as

r

用

ar^3(r-1)=576

除以

a(r-1)=9

Divide

ar^3(r-1)=576

by

a(r-1)=9

设首项为

a

，公比为

r

。由条件可得

\begin{aligned} ar^3(r-1)&=576,\\ a(r-1)&=9 \end{aligned}\text{。}

两式相除得

r^3=64

，所以

r=4

。于是

3a=9

，从而

a=3

。所求和为

\begin{aligned} S&=3(1+4+16+64+256)\\ &=3\cdot341\\ &=1023 \end{aligned}\text{。}

因此，正确答案是 B。

Let the first term be

a

and the ratio be

r.

The conditions give

\begin{aligned} ar^3(r-1)&=576,\\ a(r-1)&=9. \end{aligned}

Their quotient gives

r^3=64,

so

r=4.

Then

3a=9,

hence

a=3.

The sum is

\begin{aligned} S&=3(1+4+16+64+256)\\ &=3\cdot341\\ &=1023. \end{aligned}

Therefore, the correct answer is B.

其他年份的第 21 题