2002 AMC 12B Problems/Problem 9
Problem
If are positive real numbers such that form an increasing arithmetic sequence and form a geometric sequence, then is
Solution
Solution 1
We can let , , , and .
Solution 2
As is a geometric sequence, let and for some .
Now, is an arithmetic sequence. Its difference is . Thus .
Comparing the two expressions for we get . The positive solution is , and .
Solution 3
Letting be the common difference of the arithmetic progression, we have , , . We are given that = , or Cross-multiplying, we get So .
See also
2002 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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.