Existence of the Limit of Ratios of Consecutive Terms for a Class of Linear Recurrences

Let ( F n ) n = 1 ∞ be the classical Fibonacci sequence. It is well known that the lim F n + 1 / F n exists and equals the Golden Mean. If, more generally, ( F n ) n = 1 ∞ is an order- k linear recurrence with real constant coefficients, i.e., F n = ∑ j = 1 k λ k + 1 − j F n − j with n > k , λ j ∈ R , j = 1 , … , k , then the existence of the limit of ratios of consecutive terms may fail. In this paper, we show that the limit exists if the first k elements F 1 , F 2 , … , F k of ( F n ) n = 1 ∞ are positive, λ 1 , … , λ k − 1 are all nonnegative, at least one being positive, and max ( λ 1 , … , λ k ) = λ k ≥ k . The limit is characterized as fixed point, bounded below by λ k and bounded above by λ 1 + λ 2 + ⋯ + λ k .