2n+1/2n=42

Solution for 2n+1/2n=42 equation:

2n+1/2n=42

We move all terms to the left:

2n+1/2n-(42)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We multiply all the terms by the denominator


2n*2n-42*2n+1=0

Wy multiply elements

4n^2-84n+1=0

a = 4; b = -84; c = +1;
Δ = b2-4ac
Δ = -842-4·4·1
Δ = 7040
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{7040}=\sqrt{64*110}=\sqrt{64}*\sqrt{110}=8\sqrt{110}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-84)-8\sqrt{110}}{2*4}=\frac{84-8\sqrt{110}}{8}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-84)+8\sqrt{110}}{2*4}=\frac{84+8\sqrt{110}}{8}
$