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

视频讲解：

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文字解答：

设第一项为 $x$ ，第二项为 $y$ 。

序列前几项为因此 $x^3y^5 = 4000$ 。 $x, y, xy, xy^2, x^2y^3, x^3y^5, \cdots$

分解 $4000$ ：因为 $x$ 和 $y$ 是正整数， $4000$ 中可能作为五次方的因子只有 $1$ 或 $32$ 。 $4000 = 2^5 \cdot 5^3.$

若 $y = 1$ ，则 $x^3 = 4000$ ，但 $4000$ 不是完全立方数。因此 $y^5 = 32$ $y = 2.$

这迫使 $x$ 等于 $5$ 。

正确答案是 D。

Let $x$ be the first term and $y$ be the second.

Then we get the following sequence. $x, y, xy, xy^2, x^2y^3, x^3y^5, \cdots$ From this, we get that $x^3y^5 = 4000.$

Factoring $4000,$ we get $4000 = 2^5 \cdot 5^3.$ We need $x$ and $y$ to be integers. The only fifth powers that divides $4000$ are $1$ and $32.$

If $y = 1,$ then $x^3 = 4000,$ which doesn't work since $4000$ is not a perfect cube. Therefore, $y^5 = 32$ $y = 2.$

This forces $x$ to equal $5.$

Thus, D is the correct answer.

2023 AMC 8 第 22 题

答案：D

难度评级：1790

视频讲解：

文字解答：

其他年份的第 22 题