Given k ≥ 2, let a_n be the sequence defined by the recurrence a_n = α₁a_n–1 + … + α_ka_n–k for n ≥ k, with initial values a₀ = a₁ = … = a_k–2 = 0 and a_k–1 = 1. We show under a couple of assumptions concerning the constants α_i that the ratio $\frac{\sqrt[n]{a_n}}{\sqrt[n-1]{a_{n-1}}}$ is strictly decreasing for all n ≥ N, for some N depending on the sequence, and has limit 1. In particular, this holds in the cases when all of the α_i are unity or when all of the α_i are zero except for the first and last, which are unity. Furthermore, when k = 3 or k = 4, it is shown that one may take N to be an integer less than 12 in each of these cases.