2016 AMC 12B Problems/Problem 14
Contents
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 second term in a geometric series is , where is the common ratio for the series and is the first term of the series. So we know that and we wish to find the minimum value of the infinite sum of the series. We know that: and substituting in , we get that . From here, you can either use calculus or AM-GM.
Let , then . Since and are undefined . This means that we only need to find where the derivative equals , meaning . So , meaning that
For 2 positive real numbers and , . Let and . Then: . This implies that . or . Rearranging : . Thus, the smallest value is .
Solution 2
A geometric sequence always looks like
and they say that the second term . You should know that the sum of an infinite geometric series (denoted by here) is . We now have a system of equations which allows us to find in one variable.
We seek the smallest positive value of . We proceed by graphing in the plane (if you want to be rigorous, only stop graphing when you know all the rest you didn't graph is just the approaching of asymptotes and infinities) and find the answer is
We seek the smallest positive value of . We proceed by graphing in the plane and find the answer is
We seek the smallest positive value of . and at and . and is negative (implying a relative maximum occurs at ) and is positive (implying a relative minimum occurs at ). At , . Since this a competition math problem with an answer, we know this relative minimum must also be the absolute minimum among the "positive parts" of and that our answer is indeed However, to be sure of this outside of this cop-out, one can analyze the end behavior of , how behaves at its asymptotes, and the locations of its maxima and minima relative to the asymptotes to be sure that 4 is the absolute minimum among the "positive parts" of .
We seek the smallest positive value of . We could use calculus like we did in the solution immediately above this one, but that's a lot of work and we don't have a ton of time. To minimize the positive value of this fraction, we must maximize the denominator. The denominator is a quadratic that opens down, so its maximum occurs at its vertex. The vertex of this quadratic occurs at and .
See Also
2016 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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