F(n)=1/3n+1

Solution for F(n)=1/3n+1 equation:

(F)=1/3F+1

We move all terms to the left:

(F)-(1/3F+1)=0


Domain of the equation: 3F+1)!=0

F∈R


We get rid of parentheses

F-1/3F-1=0

We multiply all the terms by the denominator


F*3F-1*3F-1=0

Wy multiply elements

3F^2-3F-1=0

a = 3; b = -3; c = -1;
Δ = b2-4ac
Δ = -32-4·3·(-1)
Δ = 21
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-\sqrt{21}}{2*3}=\frac{3-\sqrt{21}}{6}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+\sqrt{21}}{2*3}=\frac{3+\sqrt{21}}{6}
$