Difference between revisions of "2022 AMC 12A Problems/Problem 8"
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− | ==Video Solution (HOW TO THINK CREATIVELY!!!)== | + | ==Video Solution 1 (HOW TO THINK CREATIVELY!!!)== |
https://youtu.be/_YDTIEuXTzY | https://youtu.be/_YDTIEuXTzY | ||
~Education, the Study of Everything | ~Education, the Study of Everything | ||
+ | |||
+ | ==Video Solution 2 (Smart and Fun!!!)== | ||
+ | https://youtu.be/7yAh4MtJ8a8?si=OJHbJh4_xMjBc9OY&t=1397 | ||
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+ | ~Math-X | ||
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== See Also == | == See Also == |
Revision as of 11:34, 25 October 2023
Contents
[hide]Problem
The infinite product
evaluates to a real number. What is that number?
Solution 1
We can write as
. Similarly,
.
By continuing this, we get the form
which is
Using the formula for an infinite geometric series
, we get
Thus, our answer is
.
- phuang1024
Solution 2
We can write this infinite product as (we know from the answer choices that the product must converge):
If we raise everything to the third power, we get:
Since
is positive (as it is an infinite product of positive numbers), it must be that
~ Oxymoronic15
Solution 3
Move the first term inside the second radical. We get
Do this for the third radical as well:
It is clear what the pattern is. Setting the answer as
we have
from which
~kxiang
Video Solution 1 (HOW TO THINK CREATIVELY!!!)
~Education, the Study of Everything
Video Solution 2 (Smart and Fun!!!)
https://youtu.be/7yAh4MtJ8a8?si=OJHbJh4_xMjBc9OY&t=1397
~Math-X
See Also
2022 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
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 AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.