The middle term in the expansion of `(x- (1)/(x))^(18)` is

Options

A. `""^(18)C_(9)`
B. `-""^(18)C_(9)`
C. `""^(18)C_(10)`
D. `-""^(18)C_(10)`

Answer

To find the middle term in the expansion of \((x - \frac{1}{x})^{18}\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Terms**:
The expansion of \((x - \frac{1}{x})^{18}\) will have \(n + 1\) terms, where \(n\) is the exponent. Here, \(n = 18\), so the total number of terms is \(18 + 1 = 19\).

**Hint**: Remember that the number of terms in a binomial expansion is given by \(n + 1\).

2. **Determine the Middle Term**:
Since the total number of terms (19) is odd, the middle term will be the \(\frac{19 + 1}{2} = 10^{th}\) term.

**Hint**: For an odd number of terms, the middle term is found at position \(\frac{n + 1}{2}\).

3. **Use the Binomial Theorem**:
The general term \(T_r\) in the expansion of \((a + b)^n\) is given by:
\[ T_r = \binom{n}{r} a^{n-r} b^r \]
In our case, \(a = x\), \(b = -\frac{1}{x}\), and \(n = 18\). The \(10^{th}\) term corresponds to \(r = 9\) (since we start counting from \(0\)):
\[ T_{10} = \binom{18}{9} x^{18-9} \left(-\frac{1}{x}\right)^9 \]

**Hint**: The \(r^{th}\) term corresponds to \(T_{r+1}\) in the expansion.

4. **Calculate the \(10^{th}\) Term**:
Substitute \(n\) and \(r\) into the formula:
\[ T_{10} = \binom{18}{9} x^{9} \left(-\frac{1}{x}\right)^9 \]
Simplifying this:
\[ T_{10} = \binom{18}{9} x^{9} \cdot \left(-1\right)^9 \cdot \frac{1}{x^9} \]
\[ T_{10} = \binom{18}{9} \cdot (-1) \cdot 1 = -\binom{18}{9} \]

**Hint**: Remember that \((-1)^9 = -1\).

5. **Final Result**:
Therefore, the middle term in the expansion of \((x - \frac{1}{x})^{18}\) is:
\[ -\binom{18}{9} \]

### Summary:
The middle term in the expansion of \((x - \frac{1}{x})^{18}\) is \(-\binom{18}{9}\).