  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 $F_n$.

Number Theory