F(n)=1/2n+1

Solution for F(n)=1/2n+1 equation:

(F)=1/2F+1

We move all terms to the left:

(F)-(1/2F+1)=0


Domain of the equation: 2F+1)!=0

F∈R


We get rid of parentheses

F-1/2F-1=0

We multiply all the terms by the denominator


F*2F-1*2F-1=0

Wy multiply elements

2F^2-2F-1=0

a = 2; b = -2; c = -1;
Δ = b2-4ac
Δ = -22-4·2·(-1)
Δ = 12
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{12}=\sqrt{4*3}=\sqrt{4}*\sqrt{3}=2\sqrt{3}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{3}}{2*2}=\frac{2-2\sqrt{3}}{4}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{3}}{2*2}=\frac{2+2\sqrt{3}}{4}
$