2002 AIME II Problems/Problem 3
Problem
It is given that where
and
are positive integers that form an increasing geometric sequence and
is the square of an integer. Find
Solution 1
. Since they form an increasing geometric sequence,
is the geometric mean of the product
.
.
Since is the square of an integer, we can find a few values of
that work:
and
. Out of these, the only value of
that works is
, from which we can deduce that
.
Thus,
Solution 2(similar to Solution 1)
Let be the common ratio of the geometric sequence. Since it is increasing, that means that
, and
. Simplifying the logarithm, we get
. Therefore,
. Taking the cube root of both sides, we see that
. Now since
, that means
. Using the trial and error shown in solution 1, we get
, and
.
. Therefore, the answer is
~idk12345678
See also
2002 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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