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 A as a term in the sequence. Where $a_6 = 4000 We identify by observation and a bit of multiplying for the rest of the terms in the sequence based on the given rules. We use C and D as our first 2 numbers$ a_1 = C(First Number in the Seqence) $a_2 = D ...$ a_6 = C^3 \cdot D^5 -> 4000$$ (Error compiling LaTeX. Unknown error_msg) Therefore after prime factorizing 4000 as 5^3 \cdot 2^5, We find $\boxed{\text{(D)}5}$ as our anwser ~cxrupptedpat, wuwang2002, apex304

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2023 AMC 8 Problems/Problem 22

Problem

Solution

Animated Video Solution