The geometric series a+ar+ar^2+… has a sum of 7, and the terms involving odd powers of r have a sum of 3. What is a+r?

2007 AMC 12B Problem 15

Attempt Problem 15 of the 2007 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

15.

The geometric series

a+ar+ar^2+\cdots

has a sum of

7,

and the terms involving odd powers of

r

have a sum of

3.

What is

a+r?

\dfrac{4}{3}

\dfrac{12}{7}

\dfrac{3}{2}

\dfrac{7}{3}

\dfrac{5}{2}

The even-power terms sum to

7-3=4

The odd-power series is

r

times the even-power series, so

3=r\cdot4

The odd-power terms are

ar+ar^3+\cdots

=r(a+ar^2+\cdots),

that is,

r

times the even-power terms. The even-power terms sum to

7-3=4.

3=4r,

giving

r=\tfrac34.

Then

a=7(1-r)=\tfrac74,

and

a+r=\dfrac74+\dfrac34=\dfrac52.

Thus, the correct answer is E.