I have seen the following problem in my regular study, which I am not able to solve. Please solve it for me.
Let $C_0,..., C_{k-1}$ be fixed integers and consider the recurrence relation of order $k$, $X_{n+k} = C_{k-1} X_{n+k-1} + C_{k-2} X_{n+k-2} + \cdots + C_1 X_{n+1} + C_0 X_n$. Show that once $k$ values of $X_m$ are specified, all values of $X_n$ are determined.
Let $f(r) = r^k - C_{k-1} r^{k-1} - \cdots - C_0$; we call this characteristic equation (polynomial) of the recurrence relation. Show that if $f(p) = 0$ then $X_n = cp^n$ satisfies the recurrence relation for any $c$ belongs to $\mathbb{C}$ (complex member)
Thanks in very advance... mahima