Find ` C_0 + (C_1)/(2) + (C_2)/(2^2) + (C_3)/(2^3)+…..+(C_n)/(2^n)=` ?

Options

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

Answer

To solve the problem, we need to find the sum

\[ S = C_0 + \frac{C_1}{2} + \frac{C_2}{2^2} + \frac{C_3}{2^3} + \ldots + \frac{C_n}{2^n} \]

where \( C_k \) represents the binomial coefficients from the binomial expansion of \( (1 + x)^n \).

### Step-by-step Solution:

1. **Understanding Binomial Expansion**:
The binomial theorem states that:
\[ (1 + x)^n = C_0 + C_1 x + C_2 x^2 + C_3 x^3 + \ldots + C_n x^n \]
where \( C_k = \binom{n}{k} \).

2. **Substituting \( x = \frac{1}{2} \)**:
We want to evaluate the sum \( S \) using the binomial expansion. We substitute \( x = \frac{1}{2} \) into the binomial expansion:
\[ (1 + \frac{1}{2})^n = C_0 + C_1 \cdot \frac{1}{2} + C_2 \cdot \left(\frac{1}{2}\right)^2 + C_3 \cdot \left(\frac{1}{2}\right)^3 + \ldots + C_n \cdot \left(\frac{1}{2}\right)^n \]

3. **Calculating the Left Side**:
Now calculate the left side:
\[ (1 + \frac{1}{2})^n = \left(\frac{3}{2}\right)^n \]

4. **Equating Both Sides**:
From the substitution, we have:
\[ S = C_0 + \frac{C_1}{2} + \frac{C_2}{2^2} + \frac{C_3}{2^3} + \ldots + \frac{C_n}{2^n} = \left(\frac{3}{2}\right)^n \]

5. **Final Result**:
Therefore, the required sum is:
\[ S = \left(\frac{3}{2}\right)^n \]

### Final Answer:
\[ C_0 + \frac{C_1}{2} + \frac{C_2}{2^2} + \frac{C_3}{2^3} + \ldots + \frac{C_n}{2^n} = \left(\frac{3}{2}\right)^n \]