Let's simplify the given expression step by step:
[tex]\[
\frac{2 \cdot 3^{n+1} + 7 \cdot 3^{n-1}}{3^{n+2} - 2 \left( \frac{1}{3} \right)^{1-n}}
\][/tex]
### 1. Simplify the numerator
First, look at the numerator:
[tex]\[
2 \cdot 3^{n+1} + 7 \cdot 3^{n-1}
\][/tex]
We know that:
[tex]\[
3^{n+1} = 3 \cdot 3^n
\][/tex]
[tex]\[
3^{n-1} = \frac{3^n}{3}
\][/tex]
So, substituting these back, the numerator becomes:
[tex]\[
2 \cdot 3 \cdot 3^n + 7 \cdot \frac{3^n}{3}
\][/tex]
Simplify each term:
[tex]\[
2 \cdot 3 \cdot 3^n = 6 \cdot 3^n
\][/tex]
[tex]\[
7 \cdot \frac{3^n}{3} = \frac{7}{3} \cdot 3^n = \frac{7 \cdot 3^n}{3} = \frac{7 \cdot 3^n}{3}
\][/tex]
So, the numerator can be written as:
[tex]\[
6 \cdot 3^n + \frac{7 \cdot 3^n}{3}
\][/tex]
Combine like terms:
[tex]\[
6 \cdot 3^n + \frac{7 \cdot 3^n}{3} = \frac{18 \cdot 3^n + 7 \cdot 3^n}{3} = \frac{25 \cdot 3^n}{3}
\][/tex]
### 2. Simplify the denominator
Now, look at the denominator:
[tex]\[
3^{n+2} - 2 \left( \frac{1}{3} \right)^{1-n}
\][/tex]
We know that:
[tex]\[
3^{n+2} = 3^2 \cdot 3^n = 9 \cdot 3^n
\][/tex]
[tex]\[
\left( \frac{1}{3} \right)^{1-n} = \left( 3^{-1} \right)^{1-n} = 3^{-(1-n)} = 3^{n-1}
\][/tex]
So, substituting these back, the denominator becomes:
[tex]\[
9 \cdot 3^n - 2 \cdot 3^{n-1}
\][/tex]
We can rewrite [tex]\( 3^{n-1} \)[/tex] as [tex]\( \frac{3^n}{3} \)[/tex]:
[tex]\[
9 \cdot 3^n - 2 \cdot \frac{3^n}{3}
\][/tex]
Simplify each term:
[tex]\[
9 \cdot 3^n = 9 \cdot 3^n
\][/tex]
[tex]\[
2 \cdot \frac{3^n}{3} = \frac{2 \cdot 3^n}{3} = \frac{2 \cdot 3^n}{3} = \frac{6 \cdot 3^n}{3}
\][/tex]
So, the denominator will be:
[tex]\[
9 \cdot 3^n - \frac{6 \cdot 3^n}{3}
\][/tex]
Combine like terms:
[tex]\[
9 \cdot 3^n - \frac{6 \cdot 3^n}{3} = \frac{27 \cdot 3^n - 6 \cdot 3^n}{3} = \frac{21 \cdot 3^n}{3}
\][/tex]
### 3. Simplify the fraction
The simplified fraction now looks like this:
[tex]\[
\frac{\frac{25 \cdot 3^n}{3}}{\frac{21 \cdot 3^n}{3}}
\][/tex]
We can cancel out [tex]\( 3^n \)[/tex] from the numerator and denominator:
[tex]\[
\frac{25}{21}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\frac{25}{21}
\][/tex]
