# Math Help - Roots of a quadratic

1. ## Roots of a quadratic

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 + mx - 1 = 0$, where m is an odd integer. Let $\lambda_n = \alpha^n + \beta^n$, for $n \ge 0$. Prove that for $n \ge 0$,
(a) $\lambda_n$ is an integer; and
(b) gcd $(\lambda_n, \lambda_{n + 1}) = 1$.

2. Originally Posted by alexmahone
Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 + mx - 1 = 0$, where m is an odd integer. Let $\lambda_n = \alpha^n + \beta^n$, for $n \ge 0$. Prove that for $n \ge 0$,
(a) $\lambda_n$ is an integer; and
(b) gcd $(\lambda_n, \lambda_{n + 1}) = 1$.
If $\alpha$ and $\beta$ are solutions to that equation, then $(x-\alpha)(x-\beta)= x^2+ mx-1$. Multiplying out the left side, $x^2- (\alpha+ \beta)x+ \alpha\beta= mx^2+ mx- 1$

That tells you $\alpha+ \beta= -m$ and $\alpha\beta= -1$. Does that help?