2007 AMC 12B Problem 15

Problem 15 of 25IntermediateAlgebra

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?

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Solution

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.

Concepts: geometric sequence

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.