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(1) Product and sum of the roots of the quadratic equation.

(2)
1
a
 + 1
a2
 = a+ 1
a2
 =1
as a satisfies x2=x+1. Similarly for b.

(3)
Fn + Fn+1
= 1
Ö5
 (an -bn+an+1-bn+1)
= 1
Ö5
 (an+2( 1
a
 + 1
a2
 )-bn+2( 1
b
 + 1
b2
 ))
= 1
Ö5
 (an+2-bn+2)
=Fn+2 .

(4) For n=1 the expression gives 2 = F3

For n=2 the expression gives 3=F4

For n=3 the expression gives 5 = F5

For n=4 the expression gives 8 = F6.

Conjecture : the result is Fn+2,
We have shown that this conjectureis true for n=1 ¼4.
Assume that the result is true for n=k then
1 + F1 +F2 + ¼Fk +Fk+1
= Fk+2 + Fk+1
= Fk+3
and hence the result is true for n=k+1. By the axiom of induction the result is true for all positive integer values of n.