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Sagot :
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]
[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]
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