2009 AMC 12A Problems/Problem 17
Problem
Let and be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is , and the sum of the second series is . What is ?
Solution
Using the formula for the sum of a geometric series we get that the sums of the given two sequences are and .
Hence we have and . This can be rewritten as .
As we are given that and are distinct, these must be precisely the two roots of the equation .
Using Vieta's formulas we get that the sum of these two roots is .
Solution 2
We basically have two infinite geometric series whose sum is equivalent to the common ratio. Let us have a geometric series: .
The sum is: Thus, and by Vieta's, the sum of the two possible values of ( and ) is .
~conantwiz2023
Alternate Solution
Using the formula for the sum of a geometric series we get that the sums of the given two sequences are and .
Hence we have and . This can be rewritten as .
Which can be further rewritten as . Rearranging the equation we get . Expressing this as a difference of squares we get .
Dividing by like terms we finally get as desired.
Note: It is necessary to check that , as you cannot divide by zero. As the problem states that the series are different, , and so there is no division by zero error.
See Also
2009 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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