The value of `(3xx9^(n+1)+9xx3^(2n-1))/(9xx3^(2n)-6xx9^(n-1))` is equal to

Options

A. `3(3)/(5)`
B. `3(2)/(5)`
C. `3(1)/(5)`
D. `3`

Answer

To solve the expression \((3 \cdot 9^{n+1} + 9 \cdot 3^{2n-1}) / (9 \cdot 3^{2n} - 6 \cdot 9^{n-1})\), we will first rewrite all terms in terms of base 3.

### Step 1: Rewrite the expression using base 3
We know that \(9 = 3^2\). Therefore, we can rewrite the expression as follows:

\[ 3 \cdot (3^2)^{n+1} + (3^2) \cdot 3^{2n-1} \]

This simplifies to:

\[ 3 \cdot 3^{2(n+1)} + 3^2 \cdot 3^{2n-1} \]

### Step 2: Simplify the numerator
Now we can simplify each term in the numerator:

1. The first term:
\[ 3 \cdot 3^{2(n+1)} = 3^{1 + 2(n+1)} = 3^{2n + 3} \]

2. The second term:
\[ 3^2 \cdot 3^{2n-1} = 3^{2 + (2n - 1)} = 3^{2n + 1} \]

Thus, the numerator becomes:

\[ 3^{2n + 3} + 3^{2n + 1} \]

### Step 3: Factor the numerator
We can factor out the common term \(3^{2n + 1}\):

\[ 3^{2n + 1}(3^2 + 1) = 3^{2n + 1}(9 + 1) = 3^{2n + 1} \cdot 10 \]

### Step 4: Simplify the denominator
Now, let's simplify the denominator:

\[ 9 \cdot 3^{2n} - 6 \cdot 9^{n-1} = 9 \cdot 3^{2n} - 6 \cdot (3^2)^{n-1} \]

This simplifies to:

\[ 9 \cdot 3^{2n} - 6 \cdot 3^{2(n-1)} = 9 \cdot 3^{2n} - 6 \cdot 3^{2n - 2} \]

### Step 5: Factor the denominator
We can factor out \(3^{2n - 2}\):

\[ 3^{2n - 2}(9 \cdot 3^2 - 6) = 3^{2n - 2}(9 \cdot 9 - 6) = 3^{2n - 2}(81 - 6) = 3^{2n - 2} \cdot 75 \]

### Step 6: Combine the results
Now we can substitute the simplified numerator and denominator back into the expression:

\[ \frac{3^{2n + 1} \cdot 10}{3^{2n - 2} \cdot 75} \]

### Step 7: Simplify the fraction
We can simplify this fraction by canceling out the powers of 3:

\[ \frac{10}{75} \cdot 3^{(2n + 1) - (2n - 2)} = \frac{10}{75} \cdot 3^{3} = \frac{10}{75} \cdot 27 \]

### Step 8: Final simplification
Now, simplify \(\frac{10}{75}\):

\[ \frac{10}{75} = \frac{2}{15} \]

Thus, the final expression becomes:

\[ \frac{2}{15} \cdot 27 = \frac{54}{15} = \frac{18}{5} \]

### Final Answer
The value of the expression is \(\frac{18}{5}\).

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