Difference between revisions of "2004 AMC 12A Problems/Problem 14"
(→Solution) |
m (→Solution 2) |
||
Line 16: | Line 16: | ||
<math>9r^2=29+2d</math> | <math>9r^2=29+2d</math> | ||
− | Plugging in the first equation into the second, our equation becomes <math>9r^2=29+18r-22\Longrightarrow9r^2-18r | + | Plugging in the first equation into the second, our equation becomes <math>9r^2=29+18r-22\Longrightarrow9r^2-18r+7=0</math>. By the quadratic formula, <math>r</math> can either be <math>-\frac{1}{3}</math> or <math>\frac{7}{3}</math>. If <math>r</math> is <math>-\frac{1}{3}</math>, the third term (of the geometric sequence) would be <math>1</math>, and if <math>r</math> is <math>\frac{7}{3}</math>, the third term would be <math>49</math>. Clearly the minimum possible value for the third term of the geometric sequence is <math>\boxed{\mathrm{(A)}\ 1}</math>. |
== See also == | == See also == |
Revision as of 10:04, 21 December 2016
- The following problem is from both the 2004 AMC 12A #14 and 2004 AMC 10A #18, so both problems redirect to this page.
Contents
[hide]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 1
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 .
Solution 2
Let be the common difference and be the common ratio. Then the arithmetic sequence is , , and . The geometric sequence (when expressed in terms of ) has the terms , , and . Thus, we get the following equations:
Plugging in the first equation into the second, our equation becomes . By the quadratic formula, can either be or . If is , the third term (of the geometric sequence) would be , and if is , the third term would be . Clearly the minimum possible value for the third term of the geometric sequence 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 |