The value of `(2^(n) + 2^(n -1))/(2^(n+1) -2^(n))` is equal to

Options

A. 1.5
B. `2/3`
C. 2
D. 4

Answer

To solve the expression \((2^n + 2^{n-1})/(2^{n+1} - 2^n)\), we can simplify it step-by-step.

### Step 1: Rewrite the expression
The given expression is:
\[ \frac{2^n + 2^{n-1}}{2^{n+1} - 2^n} \]

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

In the denominator, we can factor out \(2^n\):
\[ 2^{n+1} - 2^n = 2^n(2 - 1) = 2^n \cdot 1 = 2^n \]

### Step 3: Substitute back into the expression
Now we can substitute these factored forms back into the expression:
\[ \frac{2^{n-1} \cdot 3}{2^n} \]

### Step 4: Simplify the fraction
Now we can simplify the fraction:
\[ \frac{2^{n-1} \cdot 3}{2^n} = 3 \cdot \frac{2^{n-1}}{2^n} = 3 \cdot \frac{1}{2} = \frac{3}{2} \]

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

---