Find the `(r+1)`th term in the expansion of `((x)/(a)-(a)/(x))^(2n)`

Explanation

To find the \((r+1)\)th term in the expansion of \(\left(\frac{x}{a} - \frac{a}{x}\right)^{2n}\), we will use the Binomial Theorem.

### Step-by-Step Solution:

1. **Identify the Binomial Expansion**:
The Binomial Theorem states that for any integers \(n\) and any terms \(p\) and \(q\):
\[ (p + q)^n = \sum_{r=0}^{n} \binom{n}{r} p^{n-r} q^r \]
In our case, let \(p = \frac{x}{a}\) and \(q = -\frac{a}{x}\), and \(n = 2n\).

2. **Write the General Term**:
The general term \(T_r\) in the expansion can be expressed as:
\[ T_r = \binom{2n}{r} \left(\frac{x}{a}\right)^{2n-r} \left(-\frac{a}{x}\right)^r \]

3. **Simplify the General Term**:
Now we simplify \(T_r\):
\[ T_r = \binom{2n}{r} \left(\frac{x^{2n-r}}{a^{2n-r}}\right) \left(-\frac{a^r}{x^r}\right) \]
This can be rewritten as:
\[ T_r = \binom{2n}{r} (-1)^r \frac{x^{2n-r}}{a^{2n-r}} \cdot \frac{a^r}{x^r} \]
\[ = \binom{2n}{r} (-1)^r \frac{x^{2n-r}}{x^r} \cdot \frac{a^r}{a^{2n-r}} \]
\[ = \binom{2n}{r} (-1)^r \frac{x^{2n-2r}}{a^{2n-2r}} \]

4. **Find the \((r+1)\)th Term**:
The \((r+1)\)th term, denoted as \(T_{r+1}\), corresponds to \(r\) being replaced by \(r+1\):
\[ T_{r+1} = \binom{2n}{r+1} \left(-1\right)^{r+1} \frac{x^{2n-2(r+1)}}{a^{2n-2(r+1)}} \]
\[ = \binom{2n}{r+1} (-1)^{r+1} \frac{x^{2n-2r-2}}{a^{2n-2r-2}} \]

5. **Final Expression**:
Thus, the \((r+1)\)th term in the expansion of \(\left(\frac{x}{a} - \frac{a}{x}\right)^{2n}\) is:
\[ T_{r+1} = \binom{2n}{r+1} (-1)^{r+1} \frac{x^{2n-2r-2}}{a^{2n-2r-2}} \]