Problem

Source: Albanian BMO TST 2010 Question 2

Tags: inequalities, analytic geometry, conics, parabola, algebra, polynomial, Vieta



Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2-ax+1=0$ we build the sequence with $ S_{n}=x_{1}^n + x_{2}^n$. a)Prove that the sequence $ \frac{S_{n}}{S_{n+1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing. b)Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}+\cdots +\frac{S_{n}}{S_{n+1}}>n-1$