Problem:
Let a+ar1+ar12+ar13+⋯ and a+ar2+ar22+ar23+⋯ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is r1, and the sum of the second series is r2. What is r1+r2?
Answer Choices:
A. 0
B. 21
C. 1
D. 21+5
E. 2 Solution:
The sum of the first series is
1−r1a=r1
from which r12−r1+a=0, and r1=21(1±1−4a). Similarly, r2=21(1±1−4a). Because r1 and r2 must be different, r1+r2=1. Such series exist as long as 0<a<41.