2022 AMC 12A Problems/Problem 8
Contents
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 |
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