ILL 1983 Problem 38

Let $\{u_n \}$ be the sequence defined by its first two terms $u_0, u_1$ and the recursion formula $$u_{n+2 }= u_n - u_{n+1}.$$(a) Show that $u_n$ can be written in the form $u_n = \alpha a^n + \beta b^n$ , where $a, b, \alpha, \beta$ are constants independent of $n$ that have to be determined.(b) If $S_n = u_0 + u_1 + \cdots + u_n$ , prove that $S_n + u_{n-1}$ is a constant independent of $n.$ Determine this constant.