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

Options

A. (a) `(n-5)/(6)`
B. (b) `(n-4)/(5)`
C. (c) `(5)/(n-4)`
D. (d) `(6)/(n-5)`

Answer

To solve the problem step by step, we will use the binomial theorem and the information given in the question.

### Step 1: Understand the Binomial Expansion
The binomial expansion of \((a - b)^n\) can be expressed using the binomial theorem. The general term in the expansion is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} (-b)^r \]

### Step 2: Identify the Fifth and Sixth Terms
We need to find the 5th and 6th terms of the expansion:
- For the 5th term (\(T_5\)), set \(r = 4\):
\[ T_5 = \binom{n}{4} a^{n-4} (-b)^4 = \binom{n}{4} a^{n-4} b^4 \]
- For the 6th term (\(T_6\)), set \(r = 5\):
\[ T_6 = \binom{n}{5} a^{n-5} (-b)^5 = \binom{n}{5} a^{n-5} (-b^5) \]

### Step 3: 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 \(T_6\):
\[ \binom{n}{4} a^{n-4} b^4 + \binom{n}{5} a^{n-5} (-b^5) = 0 \]

### Step 4: Simplify the Equation
Rearranging the equation gives:
\[ \binom{n}{4} a^{n-4} b^4 = -\binom{n}{5} a^{n-5} b^5 \]
Dividing both sides by \(a^{n-5}\) (assuming \(a \neq 0\)) and \(b^4\) (assuming \(b \neq 0\)):
\[ \binom{n}{4} a = -\binom{n}{5} b \]

### Step 5: Express \(\frac{a}{b}\)
Rearranging gives:
\[ \frac{a}{b} = -\frac{\binom{n}{5}}{\binom{n}{4}} \]

### Step 6: Calculate the Binomial Coefficients
Using the formula for binomial coefficients:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
We have:
\[ \binom{n}{5} = \frac{n!}{5!(n-5)!}, \quad \binom{n}{4} = \frac{n!}{4!(n-4)!} \]
Thus:
\[ \frac{\binom{n}{5}}{\binom{n}{4}} = \frac{4!(n-4)!}{5!(n-5)!} = \frac{1}{5} \cdot \frac{n-4}{1} = \frac{n-4}{5} \]

### Step 7: Final Expression for \(\frac{a}{b}\)
Substituting this back into our expression for \(\frac{a}{b}\):
\[ \frac{a}{b} = -\frac{n-4}{5} \]

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