Find the 9th term in the expansion of `(a/b-b/(2a)^(2))^(12)`.

Answer

To find the 9th term in the expansion of \((\frac{a}{b} - \frac{b}{2a^2})^{12}\), we can use the Binomial Theorem. The Binomial Theorem states that:

\[ (p + q)^n = \sum_{k=0}^{n} \binom{n}{k} p^{n-k} q^k \]

Where \(\binom{n}{k}\) is the binomial coefficient.

### Step-by-Step Solution:

1. **Identify p, q, and n**:
- Here, \(p = \frac{a}{b}\), \(q = -\frac{b}{2a^2}\), and \(n = 12\).

2. **Determine the term we need**:
- We want to find the 9th term. According to the formula for the \(k\)-th term in the binomial expansion, the \(n\)-th term is given by:
\[ T_k = \binom{n}{k-1} p^{n-(k-1)} q^{k-1} \]
- For the 9th term, \(k = 9\), so we need \(T_9\):
\[ T_9 = \binom{12}{8} p^{12-8} q^8 \]

3. **Calculate the binomial coefficient**:
- Calculate \(\binom{12}{8}\):
\[ \binom{12}{8} = \binom{12}{4} = \frac{12!}{4! \cdot (12-4)!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]

4. **Calculate \(p^{12-8}\)**:
- \(p^{12-8} = p^4 = \left(\frac{a}{b}\right)^4 = \frac{a^4}{b^4}\)

5. **Calculate \(q^8\)**:
- \(q^8 = \left(-\frac{b}{2a^2}\right)^8 = \frac{b^8}{(2a^2)^8} = \frac{b^8}{256a^{16}}\)

6. **Combine everything**:
- Now substitute back into the term:
\[ T_9 = 495 \cdot \frac{a^4}{b^4} \cdot \frac{b^8}{256a^{16}} \]
- Simplifying:
\[ T_9 = \frac{495 \cdot a^4 \cdot b^8}{256 \cdot a^{16} \cdot b^4} = \frac{495}{256} \cdot \frac{b^{8-4}}{a^{16-4}} = \frac{495}{256} \cdot \frac{b^4}{a^{12}} \]

7. **Final Result**:
- The 9th term in the expansion is:
\[ T_9 = \frac{495}{256} \cdot \frac{b^4}{a^{12}} \]