Difference between revisions of "2023 AMC 8 Problems/Problem 22"
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<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 10</math> | <math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 10</math> | ||
− | ==Solution== | + | ==Solution 1== |
+ | In this solution, we will use trial and error to solve. | ||
+ | <math>4000</math> can be expressed as <math>200 \times 20</math>. We divide <math>200</math> by <math>20</math> and get <math>10</math>, divide <math>20</math> by <math>10</math> and get <math>2</math>, and divide <math>10</math> by <math>2</math> to get <math>\boxed{\textbf{(D)}\ 5}</math>. No one said that they have to be in ascending order! | ||
− | + | Solution by [[User:ILoveMath31415926535|ILoveMath31415926535]] and clarification edits by apex304 | |
− | |||
− | Solution 2 | + | ==Solution 2== |
− | + | 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:xana233|xana233]] |
− | < | + | |
− | < | + | ==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!!!)== | ||
+ | |||
+ | https://youtu.be/LAeSj372-UQ | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | ==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== | ==Animated Video Solution== | ||
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~Star League (https://starleague.us) | ~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== | ||
+ | https://youtu.be/RCYRD7OLSLc | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==Video Solution by harungurcan== | ||
+ | https://www.youtube.com/watch?v=Ki4tPSGAapU&t=1249s | ||
+ | |||
+ | ~harungurcan | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC8 box|year=2023|num-b=21|num-a=23}} | ||
+ | {{MAA Notice}} |
Latest revision as of 01:54, 21 August 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 xana233
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.