2009 AMC 12A Problem 17

Problem 17 of 25IntermediateAlgebra

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?

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Solution

For a series with first term

a

and ratio

r,

the sum is

\dfrac{a}{1 - r} = r,

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.