Difference between revisions of "2002 AMC 12B Problems/Problem 9"
Amburger66 (talk | contribs) m (→Solution 3) |
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<cmath>\frac{a + n}{a} = \frac{a + 3n}{a + n}.</cmath> | <cmath>\frac{a + n}{a} = \frac{a + 3n}{a + n}.</cmath> | ||
Cross-multiplying, we get | Cross-multiplying, we get | ||
− | <cmath>a^2 + 2an + n^2 | + | <cmath>a^2 + 2an + n^2 = a^2 + 3an</cmath> |
<cmath>n^2 = an</cmath> | <cmath>n^2 = an</cmath> | ||
<cmath>n = a</cmath> | <cmath>n = a</cmath> |
Revision as of 15:40, 22 December 2015
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 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.