Difference between revisions of "2004 AMC 12A Problems/Problem 14"
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<math>\text {(A)}\ 1 \qquad \text {(B)}\ 4 \qquad \text {(C)}\ 36 \qquad \text {(D)}\ 49 \qquad \text {(E)}\ 81</math> | <math>\text {(A)}\ 1 \qquad \text {(B)}\ 4 \qquad \text {(C)}\ 36 \qquad \text {(D)}\ 49 \qquad \text {(E)}\ 81</math> | ||
− | == Solution == | + | == Solution 1 == |
− | Let <math>d</math> be the common difference. Then <math>9, 9 + d + 2 = 11 + d, 9 + 2d + 20 = 29 + 2d</math> are the terms of the geometric progression. Since the middle term is the [[geometric mean]] of the other two terms, <math>(11+d)^2 = 9(2d+29)</math> <math>\Longrightarrow d^2 + 4d - 140</math> <math>= (d+14)(d-10) = 0</math>. The smallest possible value occurs when <math>d = -14</math>, and the third term is <math>2(-14) + 29 = 1\ \mathrm{(A)}</math>. | + | Let <math>d</math> be the common difference. Then <math>9</math>, <math>9+d+2=11+d</math>, <math>9+2d+20=29+2d</math> are the terms of the geometric progression. Since the middle term is the [[geometric mean]] of the other two terms, <math>(11+d)^2 = 9(2d+29)</math> <math>\Longrightarrow d^2 + 4d - 140</math> <math>= (d+14)(d-10) = 0</math>. The smallest possible value occurs when <math>d = -14</math>, and the third term is <math>2(-14) + 29 = 1\Rightarrow\boxed{\mathrm{(A)}\ 1}</math>. |
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+ | == Solution 2 == | ||
+ | Let <math>d</math> be the common difference and <math>r</math> be the common ratio. Then the arithmetic sequence is <math>9</math>, <math>9+d</math>, and <math>9+2d</math>. The geometric sequence (when expressed in terms of <math>d</math>) has the terms <math>9</math>, <math>11+d</math>, and <math>29+2d</math>. Thus, we get the following equations: | ||
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+ | <math>9r=11+d\Rightarrow d=9r-11</math> | ||
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+ | <math>9r^2=29+2d</math> | ||
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+ | 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 11:16, 21 July 2014
- The following problem is from both the 2004 AMC 12A #14 and 2004 AMC 10A #18, so both problems redirect to this page.
Contents
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 |