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 $xy$ , $xy^2$ , $x^2y^2$ , 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 $\boxed{\textbf{(D)}\ 5}$

~MrThinker

Solution 2: We assign the value a as a term in this sequence. $a_1->C$ $a_2->D$ $a_3->C \cdot D$ $a_4->C\cdot D^2$ $a_5->C^2 \cdot D^3$ $a_6->C^3 \cdot D^5 -> 4000$ When we prime factorize $4000, we see that$ 4000 = 2^5 \cdot 5^3$$ (Error compiling LaTeX. Unknown error_msg)

We get C=5 and D=2
Remember C is the first number so,
Our answer is

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2023 AMC 8 Problems/Problem 22

Problem

Solution

Animated Video Solution