If `(1+2x+x^(2))^(n) = sum_(r=0)^(2n)a_(r)x^(r)`, then `a_(r) = `

Options

A. (a) `(.^(n)C_(r))^(2)`
B. (b) `.^(n)C_(r)..^(n)C_(r+1)`
C. (c) `.^(2n)C_(r)`
D. (d) `.^(2n)C_(r+1)`

Explanation

To solve the problem, we need to find the coefficients \( a_r \) in the expansion of \( (1 + 2x + x^2)^n \) expressed as a sum of powers of \( x \).

### Step-by-step Solution:

1. **Rewrite the Expression**:
We start with the expression \( (1 + 2x + x^2)^n \). We can notice that \( 1 + 2x + x^2 \) can be rewritten in a different form.

\[ 1 + 2x + x^2 = (1 + x)^2 \]

Therefore, we can express our original expression as:

\[ (1 + 2x + x^2)^n = ((1 + x)^2)^n = (1 + x)^{2n} \]

2. **Expand Using Binomial Theorem**:
Now, we will expand \( (1 + x)^{2n} \) using the Binomial Theorem, which states that:

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

In our case, \( a = 1 \), \( b = x \), and \( n = 2n \):

\[ (1 + x)^{2n} = \sum_{r=0}^{2n} \binom{2n}{r} x^r \]

3. **Identify Coefficients**:
From the expansion, we can see that the coefficient of \( x^r \) in \( (1 + x)^{2n} \) is \( \binom{2n}{r} \).

Therefore, we can express the coefficients \( a_r \) as:

\[ a_r = \binom{2n}{r} \]

4. **Final Result**:
Thus, the value of \( a_r \) is given by:

\[ a_r = \binom{2n}{r} \]

### Summary:
The coefficients \( a_r \) in the expansion of \( (1 + 2x + x^2)^n \) are:

\[ a_r = \binom{2n}{r} \]