  This paper presents both a proof method and a result. The proof method
presented is particularly suitable for uniformly proving families of identities
satisfied by a family of recursive sequences. To illustrate the method, we
study the family of recursive sequences $F^{(k)}_n = \sum_{i=1}^k
F^{(k)}_{n-i}, n \ge 0, k \ge 2,$ with $n$ a parameter varying over integers,
and $k$ a parameter indexing members of the family. The main theorem states $
F^{(k)}_n = \sum_{j=1}^k P_{k,j} F^{(k)}_{n-jk},$ with $P$ a recursive triangle
satisfying the triangle recursion $P_{i,j}=2P_{i-1,j}- P_{i-1,j-1},$ with
appropriate initial conditions. The proof of the theorem exploits the fact that
characteristic polynomials of identities are divisible by the characteristic
polynomial of the recursion generating the underlying sequence.
