Given

F_{n} = \frac{1}{\sqrt{5}} (α^{n} - β^{n})

where

α

and

β

are solutions of the quadratic equation

x^{2} - x - 1 = 0

and

α > β

show that

(1) $α β = - 1$ , $α + β = 1$

(2) $\frac{1}{α} + \frac{1}{α^{2}} = \frac{1}{β} + \frac{1}{β^{2}} = 1$

(3) $F_{1} = F_{2} = 1$ and $F_{n} + F_{n + 1} = F_{n + 2}$ and hence $F_{n}$ is the $n$ th Fibonacci number and

(4) the sum $1 + F_{1} + F_{2} + \dots F_{n}$ gives another Fibonacci number.