Difference between revisions of "2023 AMC 8 Problems/Problem 22"
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Consider the first term is <math>a</math> and the second term is <math>b</math>. Then, the following term will be <math>ab</math>, <math>ab^2</math>, <math>a^2b^3</math> and <math>a^3b^5</math>. Notice that <math>4000=2^5\times 5^3</math>, then we obtain <math>a=\boxed{\textbf{(D)}\ 5}</math> and <math>b=2</math>. | Consider the first term is <math>a</math> and the second term is <math>b</math>. Then, the following term will be <math>ab</math>, <math>ab^2</math>, <math>a^2b^3</math> and <math>a^3b^5</math>. Notice that <math>4000=2^5\times 5^3</math>, then we obtain <math>a=\boxed{\textbf{(D)}\ 5}</math> and <math>b=2</math>. | ||
| − | Solution by [[User: | + | Solution by [[User:Slimeknight|Slimeknight]] |
==Video Solution by Math-X (Smart and Simple)== | ==Video Solution by Math-X (Smart and Simple)== | ||
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| + | ==Video Solution (Solve under 60 seconds!!!)== | ||
| + | https://youtu.be/6O5UXi-Jwv4?si=_Ld6okfFe3jfHzio&t=1020 | ||
| + | |||
| + | ~hsnacademy | ||
==Video Solution (THINKING CREATIVELY!!!)== | ==Video Solution (THINKING CREATIVELY!!!)== | ||
https://youtu.be/LAeSj372-UQ | https://youtu.be/LAeSj372-UQ | ||
| − | ~Education, the Study of Everything | + | ~Education, the Study of Everything |
| − | |||
==Video Solution 1 (Using Diophantine Equations)== | ==Video Solution 1 (Using Diophantine Equations)== | ||
Latest revision as of 17:06, 23 October 2024
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2
- 4 Video Solution by Math-X (Smart and Simple)
- 5 Video Solution (Solve under 60 seconds!!!)
- 6 Video Solution (THINKING CREATIVELY!!!)
- 7 Video Solution 1 (Using Diophantine Equations)
- 8 Video Solution 2 by SpreadTheMathLove
- 9 Animated Video Solution
- 10 Video Solution by Magic Square
- 11 Video Solution by Interstigation
- 12 Video Solution by WhyMath
- 13 Video Solution by harungurcan
- 14 See Also
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
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
Solution 2
Consider the first term is 
and the second term is 
. Then, the following term will be 
, 
, 
and 
. Notice that 
, then we obtain 
and 
.
Solution by Slimeknight
Video Solution by Math-X (Smart and Simple)
https://youtu.be/Ku_c1YHnLt0?si=uptT6DExGvKiatZK&t=4952 ~Math-X
Video Solution (Solve under 60 seconds!!!)
https://youtu.be/6O5UXi-Jwv4?si=_Ld6okfFe3jfHzio&t=1020
~hsnacademy
Video Solution (THINKING CREATIVELY!!!)
~Education, the Study of Everything
Video Solution 1 (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/DBqko2xATxs&t=3007
Video Solution by WhyMath
~savannahsolver
Video Solution by harungurcan
https://www.youtube.com/watch?v=Ki4tPSGAapU&t=1249s
~harungurcan
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. 
