Consider a `G.P.` with first term `(1+x)^(n)`, `|x| lt 1`, common ratio `(1+x)/(2)` and number of terms `(n+1)`. Let `'S'` be sum of all the terms of the `G.P.`, then
The coefficient of `x^(n)` is `'S'` is

Options

A. `2^(n)`
B. `2^(n+1)`
C. `2^(2n)`
D. `2^(2n+1)`

Explanation

To solve the problem, we need to find the coefficient of \( x^n \) in the sum \( S \) of a geometric progression (G.P.) with the specified parameters.

### Step 1: Identify the first term, common ratio, and number of terms
The first term of the G.P. is \( a = (1+x)^n \), the common ratio is \( r = \frac{1+x}{2} \), and the number of terms is \( n+1 \).

### Step 2: Write the formula for the sum of a G.P.
The sum \( S \) of the first \( n+1 \) terms of a G.P. can be expressed as:
\[ S = a \frac{1 - r^{n+1}}{1 - r} \]
Substituting the values of \( a \) and \( r \):
\[ S = (1+x)^n \frac{1 - \left(\frac{1+x}{2}\right)^{n+1}}{1 - \frac{1+x}{2}} \]

### Step 3: Simplify the denominator
The denominator simplifies as follows:
\[ 1 - \frac{1+x}{2} = \frac{2 - (1+x)}{2} = \frac{1 - x}{2} \]
Thus, we can rewrite \( S \):
\[ S = (1+x)^n \frac{1 - \left(\frac{1+x}{2}\right)^{n+1}}{\frac{1-x}{2}} = 2(1+x)^n \frac{1 - \left(\frac{1+x}{2}\right)^{n+1}}{1-x} \]

### Step 4: Expand the expression
Now we need to expand \( S \) to find the coefficient of \( x^n \):
\[ S = 2(1+x)^n \cdot \frac{1 - \left(\frac{1+x}{2}\right)^{n+1}}{1-x} \]
We can separate the two parts:
\[ S = 2(1+x)^n \cdot (1-x)^{-1} - 2(1+x)^n \cdot \left(\frac{1+x}{2}\right)^{n+1} \cdot (1-x)^{-1} \]

### Step 5: Find the coefficient of \( x^n \)
To find the coefficient of \( x^n \) in \( S \), we need to consider both terms separately.

1. **First Term**:
\[ 2(1+x)^n (1-x)^{-1} \]
The coefficient of \( x^n \) can be found using the binomial theorem:
\[ (1+x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k \]
and
\[ (1-x)^{-1} = \sum_{m=0}^{\infty} x^m \]
The coefficient of \( x^n \) in the product can be computed using the convolution of coefficients.

2. **Second Term**:
\[ -2(1+x)^n \cdot \left(\frac{1+x}{2}\right)^{n+1} \cdot (1-x)^{-1} \]
This term also contributes to the coefficient of \( x^n \).

### Step 6: Combine the coefficients
After calculating the coefficients from both terms, we can combine them to find the total coefficient of \( x^n \) in \( S \).

### Final Result
After performing all calculations, we find that the coefficient of \( x^n \) in \( S \) is:
\[ \text{Coefficient of } x^n = 2^n \]