Difference between revisions of "2004 AMC 12A Problems/Problem 14"
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| + | {{duplicate|[[2004 AMC 12A Problems|2004 AMC 12A #14]] and [[2004 AMC 10A Problems/Problem 18|2004 AMC 10A #18]]}} | ||
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== Problem == | == Problem == | ||
A [[sequence]] of three real numbers forms an [[arithmetic progression]] with a first term of <math>9</math>. If <math>2</math> is added to the second term and <math>20</math> is added to the third term, the three resulting numbers form a [[geometric progression]]. What is the smallest possible value for the third term in the geometric progression? | A [[sequence]] of three real numbers forms an [[arithmetic progression]] with a first term of <math>9</math>. If <math>2</math> is added to the second term and <math>20</math> is added to the third term, the three resulting numbers form a [[geometric progression]]. What is the smallest possible value for the third term in the geometric progression? | ||
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== See also == | == See also == | ||
{{AMC12 box|year=2004|ab=A|num-b=13|num-a=15}} | {{AMC12 box|year=2004|ab=A|num-b=13|num-a=15}} | ||
| + | {{AMC10 box|year=2004|ab=A|num-b=17|num-a=19}} | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
Revision as of 16:16, 5 December 2007
- The following problem is from both the 2004 AMC 12A #14 and 2004 AMC 10A #18, so both problems redirect to this page.
Problem
A sequence of three real numbers forms an arithmetic progression with a first term of 
. If 
is added to the second term and 
is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression?
Solution
Let 
be the common difference. Then 
are the terms of the geometric progression. Since the middle term is the geometric mean of the other two terms, 
. The smallest possible value occurs when 
, and the third term is 
.
See also
| 2004 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 13 |
Followed by Problem 15 |
| 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 | |
| 2004 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 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 10 Problems and Solutions | ||
