Given the formula for the

n

th Fibonacci number, namely

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

, prove that

(1) $(α + \frac{1}{α}) = - (β + \frac{1}{β}) = \sqrt{5}$ ,

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

(3) for any four consecutive Fibonacci numbers $F_{n} \dots F_{n + 3}$ the formula

$(F_{n} F_{n + 3})^{2} + (2 F_{n + 1} F_{n + 2})^{2}$

is the square of another Fibonacci number giving a Pythagorean triple.