Let $\displaystyle \alpha$ and $\displaystyle \beta$ be the roots of the quadratic equation $\displaystyle x^2 + mx - 1 = 0$, where m is an odd integer. Let $\displaystyle \lambda_n = \alpha^n + \beta^n$, for $\displaystyle n \ge 0$. Prove that for $\displaystyle n \ge 0$,

(a) $\displaystyle \lambda_n$ is an integer; and

(b) gcd $\displaystyle (\lambda_n, \lambda_{n + 1}) = 1$.