AMC8 2023

AMC8 2023 · Q22

AMC8 2023 · Q22. It mainly tests Sequences & recursion (algebra), Arithmetic misc.

In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is $4000$. What is the first term?

在一个正整数序列中，从第三个项开始，每一项都是前两项的乘积。第六项是 $4000$。第一项是多少？

(A) \ 1 \ 1

(B) \ 2 \ 2

(D) \ 5 \ 5

(E) \ 10 \ 10

Answer

Correct choice: (D)

正确答案：（D）

Solution

In this solution, we will use trial and error to solve. $4000$ can be expressed as $200 \times 20$. We divide $200$ by $20$ and get $10$, divide $20$ by $10$ and get $2$, and divide $10$ by $2$ to get $\boxed{\textbf{(D)}\ 5}$. No one said that they have to be in ascending order! Solution by ILoveMath31415926535 and clarification edits by apex304 Anonymous question here, if 5 is first term, how do you get 10? (answer to anonymous question) "No one said they had to be in ascending order!" means after 5, the term doesnt need to increase each time.

我们用试错法求解。\n$4000$ 可以表示为 $200 \times 20$。我们将 $200$ 除以 $20$ 得到 $10$，将 $20$ 除以 $10$ 得到 $2$，将 $10$ 除以 $2$ 得到 $\boxed{\textbf{(D)}\ 5}$。没人说它们必须按升序排列！\n\n由 ILoveMath31415926535 提供解答，apex304 进行澄清编辑。\n匿名问题：如果 $5$ 是第一项，怎么得到 $10$？\n（匿名问题的回答）“没人说它们必须按升序排列！”\n意思是 $5$ 之后，各项不一定每次都增大。

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