Given
Fn= 1
Ö5
 (an-bn)

where a and b are solutions of the quadratic equation x2-x-1=0 and a > b show that

(1)ab = -1, a+ b = 1

(2)
1
a
 + 1
a2
 = 1
b
 + 1
b2
 =1

(3)F1=F2=1 and Fn + Fn+1 = Fn+2 and hence Fn is the nth Fibonacci number and

(4) the sum 1 + F1 + F2 + ¼Fn gives another Fibonacci number.