Problem:
The sequence satisfies and for . Let be the least integer greater than for which is an integer. Find .
Solution:
Rearranging the given equation and taking the logarithm base of both sides yields
Successively substituting and adding the resulting equations produces . Thus the closed form for the sequence is , which is an integer only when is a positive integer power of . The least positive integer power of greater than of the form is , so .
The problems on this page are the property of the MAA's American Mathematics Competitions