MathDB

Let

a\geq 2

be a real number; with the roots

x_{1}

and

x_{2}

of the equation x^2\minus{}ax\plus{}1\equal{}0 we build the sequence with S_{n}\equal{}x_{1}^n \plus{} x_{2}^n. a)Prove that the sequence \frac{S_{n}}{S_{n\plus{}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}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1

2

Problems(1)