Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2023 AMC 8 Problems/Problem 22: Difference between revisions

← Older editNewer edit →
No edit summary
Cxrupptedpat (talk | contribs)
editor
7 edits
Line 9: Line 9:


~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>
<cmath>a_3->C \cdot D</cmath>       
<cmath>a_4->C\cdot D^2</cmath>
<cmath>a_5->C^2 \cdot D^3</cmath>
<cmath>a_6->C^3 \cdot D^5 -> 4000</cmath>
When we prime factorize <cmath>4000, we see that </cmath>4000 = 2^5 \cdot 5^3<math></math>
We get C=5 and D=2
Remember C is the first number so,
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)

Retrieved from "https://wiki.aopstest.com/wiki/index.php?title=2023_AMC_8_Problems/Problem_22&oldid=187513"