A remark on periods of periodic sequences modulo $m$

Let $\{G_n\}$ be a periodic sequence of integers modulo $m$ and let $\{SG_n\}$ be the partial sum sequence defined by $SG_n:= \sum_{k=0}^nG_k$ (mod $m$). We give a formula for the period of $\{SG_n\}$. We also show that for a generalized Fibonacci sequence $F(a,b)_n$ such that $F(a,b)_0=a$ and $F(a,b)_1=b$, we have $S^i F(a,b)_n = S^{i-1}F(a,b)_{n+2}-{n+i \choose i-2}a-{n+i \choose i-1} b$ where $S^i F(a,b)_n$ is the i-th partial sum sequence successively defined by $S^i F(a,b)_n := \sum_{k=0}^n S^{i-1}F(a,b)_k$. This is a generalized version of the well-known formula $\sum_{k=0}^n F_k = F_{n+2} -1$ of the Fibonacci sequence $F_n$.

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