Problem:
The sum of an infinite geometric series is a positive number , and the second term in the series is 1 . What is the smallest possible value of ?
Answer Choices:
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B.
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E.
Solution:
Let be the common ratio of the geometric series; then
Because , the smallest value of occurs when the value of is maximized. The graph of is a downward-opening parabola with vertex , so the smallest possible value of is . The optimal series is .
The problems on this page are the property of the MAA's American Mathematics Competitions