Difference between revisions of "2020 AIME I Problems/Problem 2"
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+ | ==Solution 2== | ||
+ | If we set <math>x=2^y</math>, we can obtain three terms of a geometric sequence through logarithm properties. The three terms are <cmath>\frac{y+1}{3}, \frac{y}{2}, y.</cmath> In a three-term geometric sequence, the middle term squared is equal to the product of the other two terms, so we obtain the following: <cmath>\frac{y^2+y}{3} = \frac{y^2}{4},</cmath> which can be solved to reveal <math>y = -4</math>. Therefore, <math>x = 2^{-4} = \frac{1}{16}</math>, so our answer is <math>\boxed{017}</math>. | ||
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+ | -molocyxu | ||
==See Also== | ==See Also== |
Revision as of 16:11, 12 March 2020
Note: Please do not post problems here until after the AIME.
Contents
Problem
There is a unique positive real number such that the three numbers , , and , in that order, form a geometric progression with positive common ratio. The number can be written as , where and are relatively prime positive integers. Find .
Solution
Since these form a geometric series, is the common ratio. Rewriting this, we get by base change formula. Therefore, the common ratio is 2. Now
. Therefore, .
~ JHawk0224
Solution 2
If we set , we can obtain three terms of a geometric sequence through logarithm properties. The three terms are In a three-term geometric sequence, the middle term squared is equal to the product of the other two terms, so we obtain the following: which can be solved to reveal . Therefore, , so our answer is .
-molocyxu
See Also
2020 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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