Problem:
Except for the first two terms, each term of the sequence is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer produces a sequence of maximum length?
Solution:
Let . The given sequence is
Except for alternating signs, the coefficients of and in this sequence appear to belong to the Fibonacci-type sequence , in which each term is the sum of its two predecessors. Because the goal is to avoid negative terms in the given sequence, an optimal satisfies as many of the following inequalities as possible before failing:
Each inequality involves a ratio of two successive terms of the Fibonacci sequence. Beginning with the fourth inequality, an optimal must satisfy
It follows that , which produces the fourteen-term sequence , .
Challenge: Prove that the coefficients of and do appear unsigned in the Fibonacci sequence.
The problems on this page are the property of the MAA's American Mathematics Competitions