3^n+2+3^n=3^n+1/7^n+1

Solution for 3^n+2+3^n=3^n+1/7^n+1 equation:

3^n+2+3^n=3^n+1/7^n+1

We move all terms to the left:

3^n+2+3^n-(3^n+1/7^n+1)=0


Domain of the equation: 7^n+1)!=0

n∈R


We get rid of parentheses

3^n+3^n-3^n-1/7^n-1+2=0

We multiply all the terms by the denominator


3^n*7^n+3^n*7^n-3^n*7^n-1*7^n+2*7^n-1=0

Wy multiply elements

21n^2+21n^2-21n^2-7n+14n-1=0

We add all the numbers together, and all the variables

21n^2+7n-1=0

a = 21; b = 7; c = -1;
Δ = b2-4ac
Δ = 72-4·21·(-1)
Δ = 133
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(7)-\sqrt{133}}{2*21}=\frac{-7-\sqrt{133}}{42}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(7)+\sqrt{133}}{2*21}=\frac{-7+\sqrt{133}}{42}
$