Problem:
In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by . Find the sum of the four terms.
Solution:
Let , and be the terms of the sequence, with and positive integers. Then , which yields . It follows that either and or and . In the first case, is or , and the second case has no solutions. When , and when . Thus, the only acceptable sequence is , and the sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions