A Property of Recurring Series

If (U): U1, U2, ... , Un,.., denotes a sequence of rational numbers satisfying a linear difference equation of order k with rational coefficients, then if a group of k consecutive terms of (U) ever repeats itself, all the roots of the polynomial associated with the difference equation are roots of unity, and the sequence (U) is periodic. [1] I show here that the like occurs, generally speaking, if one term of the sequence repeats itself at regular intervals a sufficient number of times.