Evaluate `(2^(n)+ 2^(n-1))/(2^(n+1) - 2^(n))`

Options

A. `(1)/(2)`
B. `(2)/(3)`
C. `(3)/(2)`
D. `(4)/(2)`

Answer

To evaluate the expression \((2^n + 2^{n-1}) / (2^{n+1} - 2^n)\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the original expression:
\[ \frac{2^n + 2^{n-1}}{2^{n+1} - 2^n} \]

### Step 2: Factor out common terms in the numerator
In the numerator, we can factor out \(2^{n-1}\):
\[ 2^n + 2^{n-1} = 2^{n-1}(2 + 1) = 2^{n-1} \cdot 3 \]
So the numerator becomes:
\[ 2^{n-1} \cdot 3 \]

### Step 3: Factor out common terms in the denominator
In the denominator, we can factor out \(2^n\):
\[ 2^{n+1} - 2^n = 2^n(2 - 1) = 2^n \cdot 1 \]
So the denominator simplifies to:
\[ 2^n \]

### Step 4: Rewrite the expression with factored terms
Now we can rewrite the entire expression:
\[ \frac{2^{n-1} \cdot 3}{2^n} \]

### Step 5: Simplify the expression
Now we can simplify the expression by canceling \(2^{n-1}\) in the numerator with \(2^n\) in the denominator:
\[ \frac{3}{2^{n - (n-1)}} = \frac{3}{2^1} = \frac{3}{2} \]

### Final Answer
Thus, the evaluated expression is:
\[ \frac{3}{2} \]

---