If the sum of coefficients of all even powers in the expansion of `(1 + x + x^(2) + ….x^(2n))^(2)` is 221. Then the value of n is

A. 7
B. 10
C. 11
D. 9

Options

A. 7
B. 10
C. 11
D. 9

Explanation

To solve the problem, we need to find the value of \( n \) such that the sum of the coefficients of all even powers in the expansion of \( (1 + x + x^2 + \ldots + x^{2n})^2 \) equals 221.

### Step 1: Simplify the series
The series \( 1 + x + x^2 + \ldots + x^{2n} \) is a geometric series. The sum of this series can be expressed as:
\[ S = \frac{1 - x^{2n+1}}{1 - x} \]
Thus, we can rewrite our expression as:
\[ (1 + x + x^2 + \ldots + x^{2n})^2 = \left(\frac{1 - x^{2n+1}}{1 - x}\right)^2 \]

### Step 2: Find the sum of coefficients of even powers
To find the sum of coefficients of even powers, we can use the property of substituting \( x = 1 \) and \( x = -1 \):
- Let \( S_1 = \left(\frac{1 - 1^{2n+1}}{1 - 1}\right)^2 \) (which is not valid since it leads to division by zero).
- Let \( S_2 = \left(\frac{1 - (-1)^{2n+1}}{1 + 1}\right)^2 \).

However, we can find the sum of coefficients of even powers directly by using:
\[ \text{Sum of coefficients of even powers} = \frac{S_1 + S_2}{2} \]

### Step 3: Calculate \( S_1 \) and \( S_2 \)
1. For \( x = 1 \):
\[ S_1 = (2n + 1)^2 \]
2. For \( x = -1 \):
- If \( n \) is even, \( (-1)^{2n+1} = -1 \):
\[ S_2 = \left(\frac{1 - (-1)}{2}\right)^2 = 1^2 = 1 \]
- If \( n \) is odd, \( (-1)^{2n+1} = -1 \):
\[ S_2 = \left(\frac{1 + 1}{2}\right)^2 = 1^2 = 1 \]

### Step 4: Set up the equation
Now we can set up the equation based on the given information:
\[ \frac{(2n + 1)^2 + 1}{2} = 221 \]
Multiplying both sides by 2 gives:
\[ (2n + 1)^2 + 1 = 442 \]
Subtracting 1 from both sides:
\[ (2n + 1)^2 = 441 \]
Taking the square root:
\[ 2n + 1 = 21 \quad \text{or} \quad 2n + 1 = -21 \]

### Step 5: Solve for \( n \)
1. From \( 2n + 1 = 21 \):
\[ 2n = 20 \implies n = 10 \]
2. From \( 2n + 1 = -21 \):
\[ 2n = -22 \implies n = -11 \quad \text{(not valid since } n \text{ must be non-negative)} \]

Thus, the only valid solution is:
\[ n = 10 \]

### Final Answer
The value of \( n \) is \( \boxed{10} \).