2002 AMC 12B Problems/Problem 9
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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 a=1, b=2, c=3, and d=4.
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
\[a^2 + 2an + n^2 &= a^2 + 3an\] (Error compiling LaTeX. Unknown error_msg)
So .
See also
2002 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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All AMC 12 Problems and Solutions |
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