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

Suppose the first two terms were $x$ and $y$ . Then, the next terms would be $xy$ , $xy^2$ , $x^2y^3$ , and $x^3y^5$ . Since $x^3y^5$ is the sixth term, this must be equal to $4000$ . So, $x^3y^5=4000 \Rightarrow (xy)^3y^2=4000$ . Trying out the choices, we get that $x=5$ , $y=2$ , which means that the answer is $\boxed{\textbf{(D)}\ 5}$