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 Slimeknight

= Video Solution by Pi Academy

https://youtu.be/0Fb2lOHTKJo?si=muR8lEE8byZhzvQX

Video Solution by Math-X (Smart and Simple)

https://youtu.be/Ku_c1YHnLt0?si=uptT6DExGvKiatZK&t=4952 ~Math-X

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Video Solution 1 (Using Diophantine Equations)

https://youtu.be/SwPcIZxp_gY

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=ms4agKn7lqc

Animated Video Solution

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Video Solution by Magic Square

https://youtu.be/-N46BeEKaCQ?t=2649

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https://youtu.be/DBqko2xATxs&t=3007

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Video Solution by harungurcan

https://www.youtube.com/watch?v=Ki4tPSGAapU&t=1249s

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Video Solution by Dr. David

https://youtu.be/J31l_MwfKT4

2023 AMC 8 (Problems • Answer Key • Resources)
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