Problem:
The sequence is arithmetic. The sequence is geometric. Both sequences are strictly increasing and contain only integers, and is as small as possible. What is the value of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let be an arithmetic sequence and be a geometric sequence of strictly increasing positive integers.
If the common difference of the arithmetic sequence is , then
If the common ratio of the geometric sequence is , then
Thus,
Since and are positive integers, must be divisible by .
This happens whenever leaves a remainder upon division by .
The smallest integer with this property is .
Hence,
The arithmetic sequence is
and the geometric sequence is
Both are strictly increasing sequences of positive integers, and is minimal.
Finally,
The problems on this page are the property of the MAA's American Mathematics Competitions