If the middle terms in the expression of `( x^(2) + ( 1)/( x))^(2n)` is `184756x^(10)`, then what is the value of n ?

Options

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

Answer

To solve the problem step by step, we will analyze the expression and find the value of \( n \).

### Step 1: Understand the Expression
The expression given is \( (x^2 + \frac{1}{x})^{2n} \). We need to find the middle term of this expression.

**Hint:** Identify the general form of the binomial expansion.

### Step 2: Determine the Number of Terms
In the expansion of \( (a + b)^m \), the number of terms is \( m + 1 \). Here, \( m = 2n \), so the number of terms is \( 2n + 1 \).

**Hint:** Remember that the middle term in an expansion with an odd number of terms is at position \( n + 1 \).

### Step 3: Find the Middle Term
Since \( 2n + 1 \) is odd, the middle term will be the \( (n + 1)^{th} \) term. The formula for the \( k^{th} \) term in the expansion of \( (a + b)^m \) is given by:
\[ T_k = \binom{m}{k-1} a^{m - (k-1)} b^{k-1} \]
For our case, \( a = x^2 \), \( b = \frac{1}{x} \), and \( m = 2n \). The middle term \( T_{n + 1} \) can be expressed as:
\[ T_{n + 1} = \binom{2n}{n} (x^2)^{2n - n} \left(\frac{1}{x}\right)^{n} \]

**Hint:** Substitute \( k = n + 1 \) in the formula for the \( k^{th} \) term.

### Step 4: Simplify the Middle Term
Now we simplify the middle term:
\[ T_{n + 1} = \binom{2n}{n} (x^2)^{n} \left(\frac{1}{x}\right)^{n} = \binom{2n}{n} x^{2n - n} = \binom{2n}{n} x^{n} \]

**Hint:** Combine the powers of \( x \) carefully.

### Step 5: Set Up the Equation
We know from the problem statement that this middle term equals \( 184756 x^{10} \):
\[ \binom{2n}{n} x^{n} = 184756 x^{10} \]

**Hint:** Compare the coefficients and the powers of \( x \).

### Step 6: Equate the Powers of \( x \)
From the equation, we can equate the powers of \( x \):
\[ n = 10 \]

**Hint:** This gives us the value of \( n \).

### Step 7: Find the Coefficient
Now, we need to check the coefficient:
\[ \binom{2n}{n} = 184756 \]
Substituting \( n = 10 \):
\[ \binom{20}{10} = 184756 \]

**Hint:** Verify this by calculating \( \binom{20}{10} \).

### Conclusion
Thus, the value of \( n \) is:
\[ \boxed{10} \]