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−r1​a​=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​.