2009 AMC 12A Problem 17

Below is the professionally curated solution for Problem 17 of the 2009 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12A solutions, or check the answer key.

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Concepts:geometric sequence Vieta’s Formulas

Difficulty rating: 2040

17.

Let $a + ar_1 + ar_1^2 + ar_1^3 + \cdots$ and $a + ar_2 + ar_2^2 + ar_2^3 + \cdots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $r_1,$ and the sum of the second series is $r_2.$ What is $r_1 + r_2?$

$0$

$\dfrac{1}{2}$

$1$

$\dfrac{1 + \sqrt{5}}{2}$

$2$

Solution:

For a series with first term $a$ and ratio $r,$ the sum is $\dfrac{a}{1 - r} = r,$ so $r^2 - r + a = 0.$

Both $r_1$ and $r_2$ satisfy this same quadratic, and since the two series are different, $r_1 \ne r_2,$ so they are its two distinct roots. By Vieta's formulas, $r_1 + r_2 = 1.$

Thus, the correct answer is C.

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