Difference between revisions of "2003 AMC 10B Problems/Problem 8"
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The second and fourth terms of a geometric sequence are <math>2</math> and <math>6</math>. Which of the following is a possible first term? | The second and fourth terms of a geometric sequence are <math>2</math> and <math>6</math>. Which of the following is a possible first term? | ||
− | + | <math>\textbf{(A) } -\sqrt{3} \qquad\textbf{(B) } -\frac{2\sqrt{3}}{3} \qquad\textbf{(C) } -\frac{\sqrt{3}}{3} \qquad\textbf{(D) } \sqrt{3} \qquad\textbf{(E) } 3</math> | |
==Solution== | ==Solution== |
Latest revision as of 19:44, 27 March 2023
- The following problem is from both the 2003 AMC 12B #6 and 2003 AMC 10B #8, so both problems redirect to this page.
Problem
The second and fourth terms of a geometric sequence are and . Which of the following is a possible first term?
Solution
Let the first term be and the common ratio be . Therefore,
Dividing by eliminates the , yielding , so .
Now, since , , so .
We therefore see that is a possible first term.
See Also
2003 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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 |
2003 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
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.