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Difference between revisions of "2023 AMC 8 Problems/Problem 22"
(→‎Solution)
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~MrThinker
 
~MrThinker
 +
 +
Solution 2:
 +
We assign the value a as a term in this sequence.
 +
<cmath>a_1->C</cmath>
 +
<cmath>a_2->D</cmath>
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<cmath>a_3->C \cdot D</cmath>       
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<cmath>a_4->C\cdot D^2</cmath>
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<cmath>a_5->C^2 \cdot D^3</cmath>
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<cmath>a_6->C^3 \cdot D^5 -> 4000</cmath>
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When we prime factorize <cmath>4000, we see that </cmath>4000 = 2^5 \cdot 5^3<math></math>
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We get C=5 and D=2
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Remember C is the first number so,
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Our answer is <math>\boxed{\text{(D)}5}</math>
  
 
==Animated Video Solution==
 
==Animated Video Solution==

Revision as of 18:43, 24 January 2023

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 $\boxed{\text{(D)}5}$

Animated Video Solution

https://youtu.be/tnv1XzSOagA

~Star League (https://starleague.us)

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