Find the middle in the expansion of :
(i) `(x/3 + 9y)^10`
(ii) ` (x/y + y/x)^(2n+1)`

Answer

To find the middle term in the expansions of the given expressions, we will follow the Binomial Theorem, which states that:

\[ (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \]

### (i) For the expression \((\frac{x}{3} + 9y)^{10}\):

1. **Identify \(n\)**: Here, \(n = 10\), which is even.
2. **Find the middle term**: The middle term when \(n\) is even is given by the formula:
\[ \text{Middle term} = T_{\frac{n}{2} + 1} = T_{6} \]
3. **Calculate \(T_6\)**:
\[ T_6 = \binom{10}{5} \left(\frac{x}{3}\right)^{10-5} (9y)^5 \]
\[ = \binom{10}{5} \left(\frac{x}{3}\right)^5 (9y)^5 \]
4. **Calculate \(\binom{10}{5}\)**:
\[ \binom{10}{5} = \frac{10!}{5!5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 \]
5. **Substitute values**:
\[ T_6 = 252 \left(\frac{x^5}{3^5}\right) (9^5 y^5) \]
\[ = 252 \left(\frac{x^5}{243}\right) (59049 y^5) \]
6. **Final calculation**:
\[ = 252 \times \frac{59049}{243} x^5 y^5 \]
\[ = 252 \times 243 x^5 y^5 \]
\[ = 61236 x^5 y^5 \]

### (ii) For the expression \((\frac{x}{y} + \frac{y}{x})^{2n+1}\):

1. **Identify \(n\)**: Here, \(n = 2n + 1\), which is odd.
2. **Find the middle terms**: The middle terms when \(n\) is odd are given by:
\[ T_{n+1} \text{ and } T_{n+2} \]
3. **Calculate \(T_{n+1}\)**:
\[ T_{n+1} = \binom{2n+1}{n} \left(\frac{x}{y}\right)^n \left(\frac{y}{x}\right)^{(2n+1)-n} \]
\[ = \binom{2n+1}{n} \left(\frac{x}{y}\right)^n \left(\frac{y}{x}\right)^{n+1} \]
\[ = \binom{2n+1}{n} \frac{x^n y^{n+1}}{y^n x^{n+1}} = \binom{2n+1}{n} \frac{y}{x} \]
4. **Calculate \(T_{n+2}\)**:
\[ T_{n+2} = \binom{2n+1}{n+1} \left(\frac{x}{y}\right)^{n+1} \left(\frac{y}{x}\right)^{(2n+1)-(n+1)} \]
\[ = \binom{2n+1}{n+1} \left(\frac{x}{y}\right)^{n+1} \left(\frac{y}{x}\right)^{n} \]
\[ = \binom{2n+1}{n+1} \frac{x^{n+1} y^{n}}{y^{n+1} x^{n}} = \binom{2n+1}{n+1} \frac{x}{y} \]

### Final Results:
1. The middle term of \((\frac{x}{3} + 9y)^{10}\) is \(61236 x^5 y^5\).
2. The middle terms of \((\frac{x}{y} + \frac{y}{x})^{2n+1}\) are \(\binom{2n+1}{n} \frac{y}{x}\) and \(\binom{2n+1}{n+1} \frac{x}{y}\).