2023 AMC 8 第 22 题

Question

在一个正整数序列中，从第三项开始，每一项都是前两项的乘积。该序列的第六项是

4000

。第一项是多少？

In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term in the sequence is

4000.

What is the first term?

1

2

4

5

10

答案：D

知识点：递推质因数分解

难度评级：1790

小提示：

用第一项

x

和第二项

y

写出前几项

Write the first several terms using first term

x

and second term

y

大提示：

第六项是

x^3y^5

The sixth term is

x^3y^5

视频讲解：

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文字解答：

设第一项为

x

，第二项为

y

。

于是，序列的前几项为

x, y, xy, xy^2, x^2y^3, x^3y^5, \cdots

因此

x^3y^5 = 4000

。

将

4000

分解质因数，得到

4000 = 2^5 \cdot 5^3\text{。}

因为

x

和

y

都是正整数，所以能整除

4000

的完全五次方只有

1

和

32

。

若

y = 1

，则

x^3 = 4000

，但

4000

不是完全立方数，不可能成立。因此

y^5 = 32

且

y = 2\text{。}

这迫使

x

等于

5

。

所以正确答案是 D。

Let

x

be the first term and

y

be the second.

Then we get the following sequence:

x, y, xy, xy^2, x^2y^3, x^3y^5, \cdots

Thus

x^3y^5 = 4000.

Factoring

4000,

gives

4000 = 2^5 \cdot 5^3.

Because

x

and

y

are positive integers, the only fifth powers that divide

4000

are

1

and

32.

If

y = 1,

then

x^3 = 4000,

which is impossible because

4000

is not a perfect cube. Therefore,

y^5 = 32

and

y = 2.

This forces

x

to equal

5.

Thus, D is the correct answer.