F(n)=1/2n+2

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

(F)=1/2F+2

We move all terms to the left:

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


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

F∈R


We get rid of parentheses

F-1/2F-2=0

We multiply all the terms by the denominator


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

Wy multiply elements

2F^2-4F-1=0

a = 2; b = -4; c = -1;
Δ = b2-4ac
Δ = -42-4·2·(-1)
Δ = 24
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:

$\sqrt{\Delta}=\sqrt{24}=\sqrt{4*6}=\sqrt{4}*\sqrt{6}=2\sqrt{6}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{6}}{2*2}=\frac{4-2\sqrt{6}}{4}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{6}}{2*2}=\frac{4+2\sqrt{6}}{4}
$