2009 AMC 12A Problem 17

Question

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

The sum of an infinite geometric series with first term

a

and ratio

r

is

\dfrac{a}{1 - r}

Each ratio satisfies

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

that is

r^2 - r + a = 0;

use Vieta on the two roots

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.

Problem 17 in other years