2016 AMC 10B Problems/Problem 16
Problem
The sum of an infinite geometric series is a positive number , and the second term in the series is
. What is the smallest possible value of
Solution
The sum of an infinite geometric series is of the form:
where
is the first term and
is the ratio whose absolute value is less than 1.
We know that the second term(
) is the first term multiplied by the ratio.
In other words:
,
(given),
, and
.
Thus the sum is the following:
.
We can multiply
to both sides of the numerator and denominator.
.
Since we want the minimum value of this expression, we want the maximum value for the denominator which is a quadratic of the form
.
The maximum value of a quadratic with negative
is
.
.
Plugging 1/2 in, we get:
,
.
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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 |
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