2020 AIME I Problems/Problem 2
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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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