Problem: Which of the following is satisfied by all numbers x of the form
x=3a1​​+32a2​​+…+325a25​​
where a1​ is 0 or 2,a2​ is 0 or 2,…,a25​ is 0 or 2?
Answer Choices:
A. 0⩽x<1/3
B. 1/3⩽x<2/3
C. 2/3⩽x<1
D. 0⩽x<1/3 or 2/3⩽x<1
E. 1/2⩽x⩽3/4
Solution:
Using the formula for the sum of a geometric series
0≤x<n=1∑∞​(32​)n=1−31​32​​=1
If a1​=0, then
0≤x<n=2∑∞​(32​)n=1−31​92​​=31​
If α1​=2, then
32​+0+0+⋯≤x<1
So either 0≤x<31​ or 32​≤x<1.