`2xx(16xx2^(n+1)-4xx2^(n))/(16xx2^(n+2)-2xx2^(n+2))=?`

Options

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

Answer

To solve the expression \( \frac{2 \cdot 16 \cdot 2^{n+1} - 4 \cdot 2^n}{16 \cdot 2^{n+2} - 2 \cdot 2^{n+2}} \), we will simplify both the numerator and the denominator step by step.

### Step 1: Simplify the Numerator
The numerator is \( 2 \cdot 16 \cdot 2^{n+1} - 4 \cdot 2^n \).

1. Rewrite \( 4 \) as \( 2^2 \):
\[ 2 \cdot 16 \cdot 2^{n+1} - 2^2 \cdot 2^n = 2 \cdot 16 \cdot 2^{n+1} - 2^{n+2} \]

2. Factor out \( 2^{n+1} \) from the numerator:
\[ = 2^{n+1} \left( 2 \cdot 16 - 2 \right) \]
\[ = 2^{n+1} \left( 32 - 2 \right) = 2^{n+1} \cdot 30 \]

### Step 2: Simplify the Denominator
The denominator is \( 16 \cdot 2^{n+2} - 2 \cdot 2^{n+2} \).

1. Factor out \( 2^{n+2} \):
\[ = 2^{n+2} (16 - 2) = 2^{n+2} \cdot 14 \]

### Step 3: Combine the Simplified Parts
Now we can rewrite the original expression with the simplified numerator and denominator:
\[ \frac{2^{n+1} \cdot 30}{2^{n+2} \cdot 14} \]

### Step 4: Simplify the Fraction
1. Cancel \( 2^{n+1} \) from the numerator and \( 2^{n+2} \) from the denominator:
\[ = \frac{30}{14 \cdot 2} = \frac{30}{28} \]

2. Simplify \( \frac{30}{28} \):
\[ = \frac{15}{14} \]

### Final Answer
Thus, the final answer is:
\[ \frac{15}{14} \]