If the middle term in the expansion of `(x^(2)+1/x)^(2n)` is `184756x^(10)`, then the value of in is

Options

A. 10
B. 8
C. 5
D. 4

Answer

To solve the problem, we need to find the value of \( n \) such that the middle term in the expansion of \( (x^2 + \frac{1}{x})^{2n} \) is equal to \( 184756 x^{10} \).

### Step-by-Step Solution:

1. **Identify the Middle Term**:
The expansion of \( (x^2 + \frac{1}{x})^{2n} \) has \( 2n + 1 \) terms. The middle term occurs at position \( n + 1 \) (or \( r = n \)) in the expansion.

2. **Use the Binomial Theorem**:
The general term in the expansion of \( (a + b)^m \) is given by:
\[ T_r = \binom{m}{r} a^{m-r} b^r \]
For our case, \( a = x^2 \), \( b = \frac{1}{x} \), and \( m = 2n \). Thus, the middle term (when \( r = n \)) is:
\[ T_n = \binom{2n}{n} (x^2)^{2n-n} \left(\frac{1}{x}\right)^n \]

3. **Simplify the Middle Term**:
This simplifies to:
\[ T_n = \binom{2n}{n} (x^2)^{n} \left(\frac{1}{x}\right)^n = \binom{2n}{n} x^{2n-n} x^{-n} = \binom{2n}{n} x^{n} \]

4. **Set the Middle Term Equal to Given Value**:
We know that this middle term is equal to \( 184756 x^{10} \):
\[ \binom{2n}{n} x^{n} = 184756 x^{10} \]

5. **Compare Powers of \( x \)**:
From the equation \( x^{n} = x^{10} \), we can equate the powers:
\[ n = 10 \]

6. **Find the Coefficient**:
Now we need to find \( \binom{2n}{n} \) when \( n = 10 \):
\[ \binom{20}{10} \]
We can calculate \( \binom{20}{10} \) using the formula:
\[ \binom{20}{10} = \frac{20!}{10! \cdot 10!} = 184756 \]

7. **Conclusion**:
Since both the coefficient \( \binom{20}{10} \) and the power of \( x \) match the given conditions, we conclude that:
\[ n = 10 \]

### Final Answer:
The value of \( n \) is \( 10 \).