2023 AMC 8 Problems/Problem 22
Contents
Problem
In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is . What is the first term?
Solution 1
Suppose the first terms were
and
. Then, the next proceeding terms would be
,
,
, and
. Since
is the
th term, this must be equal to
. So,
. If we prime factorize
we get
. We conclude
and
, which means that the answer is
~MrThinker, numerophile (edits apex304)
Solution 2
In this solution, we will use trial and error to solve.
can be expressed as
. We divide
by
and get
, divide
by
and get
, and divide
by
to get
. No one said that they have to be in ascending order!
Solution by ILoveMath31415926535 and clarification edits by apex304
Video Solution 1 by OmegaLearn (Using Diophantine Equations)
Video Solution 2 by SpreadTheMathLove
https://www.youtube.com/watch?v=ms4agKn7lqc
Animated Video Solution
~Star League (https://starleague.us)
Video Solution by Magic Square
https://youtu.be/-N46BeEKaCQ?t=2649
Video Solution by Interstigation
https://youtu.be/1bA7fD7Lg54?t=2257
Video Solution by WhyMath
~savannahsolver
See Also
2023 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.