In the binomial expansion of `(a - b)^n , n ge 5 ` the sum of the 5th and 6th term is zero , then find `a/b`

Answer

To solve the problem, we need to find the value of \( \frac{a}{b} \) given that the sum of the 5th and 6th terms in the binomial expansion of \( (a - b)^n \) is zero.

### Step-by-Step Solution:

1. **Identify the Terms**:
In the binomial expansion of \( (a - b)^n \), the general term \( T_k \) is given by:
\[ T_k = \binom{n}{k-1} a^{n-(k-1)} (-b)^{k-1} \]
Therefore, the 5th term \( T_5 \) and the 6th term \( T_6 \) can be expressed as:
\[ T_5 = \binom{n}{4} a^{n-4} (-b)^4 = \binom{n}{4} a^{n-4} b^4 \]
\[ T_6 = \binom{n}{5} a^{n-5} (-b)^5 = -\binom{n}{5} a^{n-5} b^5 \]

2. **Set Up the Equation**:
According to the problem, the sum of the 5th and 6th terms is zero:
\[ T_5 + T_6 = 0 \]
Substituting the expressions for \( T_5 \) and \( T_6 \):
\[ \binom{n}{4} a^{n-4} b^4 - \binom{n}{5} a^{n-5} b^5 = 0 \]

3. **Rearranging the Equation**:
Rearranging gives:
\[ \binom{n}{4} a^{n-4} b^4 = \binom{n}{5} a^{n-5} b^5 \]

4. **Dividing Both Sides**:
Dividing both sides by \( a^{n-5} b^4 \) (assuming \( a \neq 0 \) and \( b \neq 0 \)):
\[ \frac{\binom{n}{4}}{\binom{n}{5}} = \frac{b}{a} \]

5. **Using the Binomial Coefficient Relation**:
The relation between the binomial coefficients is:
\[ \frac{\binom{n}{5}}{\binom{n}{4}} = \frac{n-4}{5} \]
Therefore, we have:
\[ \frac{b}{a} = \frac{5}{n-4} \]

6. **Finding \( \frac{a}{b} \)**:
Taking the reciprocal gives:
\[ \frac{a}{b} = \frac{n-4}{5} \]

### Final Answer:
Thus, the value of \( \frac{a}{b} \) is:
\[ \frac{a}{b} = \frac{n-4}{5} \]