Question 63
Let $a \geq 2$ be a real number. Denote by $x_{1}$ and $x_{2}$ the roots of the equation $x^{2}-a x+1=0$ and set $S_{n}=x_{1}^{n}+x_{2}^{n}$, $n=1,2, \ldots$
a) Prove that the sequence $\left\{\frac{S_{n}}{S_{n+1}}\right\}_{n=1}^{\infty}$ is decreasing.
b) Find all $a$ such that
$$
\frac{S_{1}}{S_{2}}+\frac{S_{2}}{S_{3}}+\cdots+\frac{S_{n}}{S_{n+1}}\gt n-1
$$
for any $n=1,2, \ldots$.
a) Prove that the sequence $\left\{\frac{S_{n}}{S_{n+1}}\right\}_{n=1}^{\infty}$ is decreasing.
b) Find all $a$ such that
$$
\frac{S_{1}}{S_{2}}+\frac{S_{2}}{S_{3}}+\cdots+\frac{S_{n}}{S_{n+1}}\gt n-1
$$
for any $n=1,2, \ldots$.