# 2023 AMC 8 Problems/Problem 22

## Problem

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?

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 10$

## Solution 1

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

## Solution 2

Consider the first term is $a$ and the second term is $b$. Then, the following term will be $ab$, $ab^2$, $a^2b^3$ and $a^3b^5$. Notice that $4000=2^5\times 5^3$, then we obtain $a=\boxed{\textbf{(D)}\ 5}$ and $b=2$.

Solution by xana233

## Video Solution (THINKING CREATIVELY!!!)

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## Animated Video Solution

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