Problem 5 · Geometric series with overlapping sets · 4 points
Made by AI and not reviewed by a real person.
Cups numbered 1, 2, 3, ... hold 2, 1, 1/2, 1/4, ... in order. Ada marks every even-numbered cup, and Ben marks every cup numbered by a multiple of 3; what is the sum in cups marked by exactly one of them?
- 4/7
- 16/9
- 116/63
- 4/3
- 40/21
Show a hint
- Find the total Ada marks and the total Ben marks separately, then think about the cups they both mark.
- The shared cups are numbered by multiples of 6. Each of the three groups has a geometric sum.
Show the answer and solution
Answer: B — 16/9
The sums in Ada’s and Ben’s cups are 1 + 1/4 + ... = 4/3 and 1/2 + 1/16 + ... = 4/7. Cups they both mark have numbers divisible by 6 and sum to 1/16 + 1/1024 + ... = 4/63. To keep only cups marked by exactly one person, subtract that shared sum twice: 4/3 + 4/7 − 2(4/63) = 16/9. Subtracting it just once gives 116/63, which incorrectly includes the shared cups.