Sequences Generated by Powers of the kth-order Fibonacci Recurrence Relation

Abstract

Every positive sequence generated by the Fibonacci recurrence relation diverges to infinity. However, every positive sequence generated by the square root of the Fibonacci recurrence relation converges to 2. In this note, we investigate different powers of the Fibonacci recurrence relation and higher-order Fibonacci recurrence relations. We then prove a theorem that gives the limit of positive sequences generated by the pth-power of the kth-order Fibonacci recurrence relation.

MSC:

• Primary 39A12

ACKNOWLEDGMENT

The authors thank the editor and the referees for their suggestions for improving the paper.

100 Years Ago This Month in The American Mathematical Monthly Edited by Vadim Ponomarenko

Professor C. A. Waldo, who retired last June from the Thayer professorship of mathematics and applied mechanics at Washington University, St. Louis, Mo., is now living at 401 West 118th Street, New York City.

In an article entitled “Medicine and mathematics in the sixteenth century” which appeared in the Annals of Medical History, summer number, 1917, Professor D. E. Smith considers the reasons for the close relations of these two branches then existing and lists a large number of men who were distinguished in both, among whom may be cited Leonardo da Vinci, Copernicus, Cardan, Gemma Frisius, and Robert Recorde.

—Excerpted from “Notes and News” 25 (1918) 238–240.