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

答案：B

知识点：等比数列求和

难度评级：1900

小提示：

设首项为 $a$ ，公比为 $r$

Write the first term as $a$ and the common ratio as $r$

大提示：

用 $ar^3(r-1)=576$ 除以 $a(r-1)=9$

Divide $ar^3(r-1)=576$ by $a(r-1)=9$

解答：

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.

1974 AMC 12 第 21 题

答案：B

难度评级：1900

小提示：

大提示：

解答：

其他年份的第 21 题