2016 AMC 10B Problems/Problem 16
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
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:
Thus, the sum is the following:
Since we want the minimum value of this expression, we want the maximum value for the denominator, . The maximum x-value of a quadratic with leading coefficient is .
Plugging into the quadratic yields:
Therefore, the minimum sum of our infinite geometric sequence is . (Solution by akaashp11)
As an extension to find the maximum value for the denominator we can find the derivative of to get . we know that this changes sign when so plugging it in into the original equation we find the answer is .
After observation we realize that in order to minimize our sum with being the reciprocal of r. The common ratio has to be in the form of with being an integer as anything more than divided by would give a larger sum than a ratio in the form of .
The first term has to be , so then in order to minimize the sum, we have minimize .
The smallest possible value for such that it is an integer that's greater than is . So our first term is and our common ratio is . Thus the sum is or . Solution 2 by No_One
We can see that if is the first term, and is the common ratio between each of the terms, then we can get Also, we know that the second term can be expressed as notice if we multiply by , we would get This quadratic has real solutions if the discriminant is greater than or equal to zero, or This yields that or . However, since we know that has to be positive, we can safely conclude that the minimum possible value of is .
Solution 4 (Quick Method)
Let the first term of the geometric series . Since it must be decreasing, we have and the third term is . Realize that by AM-GM inequality with equality if . However, we established that so that means . So the sum of the first three terms of the sequence is greater than , and the geometric series keeps continuing infinitely. This means the sum continues increasing. The only answer choice greater than is . ~skyscraper
Solution 5 (Clever Algebra)
Let the first term be The sum of the series is Rewrite this as By AM-GM we know that so the minimum is
Solution 6 (Calculus)
Set the first term is , the common ratio should be
The sum to infinity of the series is
Since is positive, we have Define the function , the domain of this function is
Let We solve that
It's easy to find that when when Thus has a minimum value when , which is Choose
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