The coefficient of `x^(4)` in the expansion of `(1+5x+9x^(2)+13x^(3)+17x^(4)+…..)(1+x^(2))^(11)` is equal to

Options

A. `.^(11)C_(2)+4.^(11)C_(1)+3`
B. `.^(11)C_(2)+3.^(11)C_(1)+4`
C. `3.^(11)C_(2)+4.^(11)C_(1)+3`
D. 171

Answer

To find the coefficient of \( x^4 \) in the expansion of \( (1 + 5x + 9x^2 + 13x^3 + 17x^4 + \ldots)(1 + x^2)^{11} \), we will follow these steps:

### Step 1: Identify the series expansion
The first part of the expression is a polynomial series:
\[ 1 + 5x + 9x^2 + 13x^3 + 17x^4 + \ldots \]
This series can be observed as a pattern where the coefficients of \( x^n \) seem to follow a linear pattern. The coefficients are \( 1, 5, 9, 13, 17 \), which can be expressed as:
\[ a_n = 1 + 4n \quad \text{for } n = 0, 1, 2, 3, 4, \ldots \]

### Step 2: Expand \( (1 + x^2)^{11} \)
Using the binomial theorem, we can expand \( (1 + x^2)^{11} \):
\[ (1 + x^2)^{11} = \sum_{k=0}^{11} \binom{11}{k} (x^2)^k = \sum_{k=0}^{11} \binom{11}{k} x^{2k} \]

### Step 3: Identify terms contributing to \( x^4 \)
We need to find combinations of terms from both expansions that will result in \( x^4 \):
1. From the first polynomial, the term \( 1 \) contributes to \( x^4 \) when multiplied by the \( x^4 \) term from \( (1 + x^2)^{11} \).
2. The \( 5x \) term cannot contribute to \( x^4 \) since it would require \( x^3 \) from \( (1 + x^2)^{11} \), which does not exist.
3. The \( 9x^2 \) term contributes to \( x^4 \) when multiplied by the \( x^2 \) term from \( (1 + x^2)^{11} \).
4. The \( 13x^3 \) term cannot contribute to \( x^4 \) since it would require \( x^1 \) from \( (1 + x^2)^{11} \), which does not exist.
5. The \( 17x^4 \) term contributes to \( x^4 \) when multiplied by the constant term \( 1 \) from \( (1 + x^2)^{11} \).

### Step 4: Calculate coefficients
Now we calculate the coefficients for each contributing term:
1. For the term \( 1 \cdot x^4 \):
- Coefficient from \( (1 + x^2)^{11} \) for \( x^4 \) is \( \binom{11}{2} \):
\[ \binom{11}{2} = \frac{11 \times 10}{2} = 55 \]

2. For the term \( 9x^2 \cdot x^2 \):
- Coefficient from \( (1 + x^2)^{11} \) for \( x^2 \) is \( \binom{11}{1} \):
\[ \binom{11}{1} = 11 \]
- Contribution: \( 9 \cdot 11 = 99 \)

3. For the term \( 17x^4 \cdot 1 \):
- Coefficient from \( (1 + x^2)^{11} \) for \( 1 \) is \( \binom{11}{0} = 1 \):
- Contribution: \( 17 \cdot 1 = 17 \)

### Step 5: Sum the contributions
Now, we sum all contributions to find the total coefficient of \( x^4 \):
\[ \text{Total Coefficient} = 55 + 99 + 17 = 171 \]

### Final Answer
The coefficient of \( x^4 \) in the expansion is \( \boxed{171} \).