Consider the expansion of `( 1+ x)^(2n+1)`
The expansion of `( x-y)^(n), n ge 5` is doen in the descending powers of x. If the sum of the fifth and sixth terms is zero, then `(x)/( y )` is equal to `:`

Options

A. `( n - 5)/( 6)`
B. `( n -4)/( 5)`
C. `( 5)/( n-4)`
D. `( 6)/( n-5)`

Answer

To solve the problem, we need to analyze the expansion of \((x - y)^n\) and find the ratio \(\frac{x}{y}\) given that the sum of the fifth and sixth terms is zero.

### Step-by-Step Solution:

1. **Identify the Terms**:
The general term \(T_r\) in the expansion of \((x - y)^n\) is given by:
\[ T_r = \binom{n}{r} x^{n-r} (-y)^r = \binom{n}{r} x^{n-r} (-1)^r y^r \]
For the fifth term (\(T_5\)), we have \(r = 4\) and for the sixth term (\(T_6\)), we have \(r = 5\).

2. **Write the Fifth and Sixth Terms**:
- Fifth Term (\(T_5\)):
\[ T_5 = \binom{n}{4} x^{n-4} (-y)^4 = \binom{n}{4} x^{n-4} y^4 \]
- Sixth Term (\(T_6\)):
\[ T_6 = \binom{n}{5} x^{n-5} (-y)^5 = -\binom{n}{5} x^{n-5} y^5 \]

3. **Set Up the Equation**:
According to the problem, the sum of the fifth and sixth terms is zero:
\[ T_5 + T_6 = 0 \]
This gives us:
\[ \binom{n}{4} x^{n-4} y^4 - \binom{n}{5} x^{n-5} y^5 = 0 \]

4. **Factor Out Common Terms**:
We can factor out \(x^{n-5} y^4\) from the equation:
\[ x^{n-5} y^4 \left( \binom{n}{4} x - \binom{n}{5} y \right) = 0 \]
Since \(x^{n-5} y^4 \neq 0\) (as \(n \geq 5\)), we have:
\[ \binom{n}{4} x - \binom{n}{5} y = 0 \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ \binom{n}{4} x = \binom{n}{5} y \]
Dividing both sides by \(y\) and \(\binom{n}{4}\):
\[ \frac{x}{y} = \frac{\binom{n}{5}}{\binom{n}{4}} \]

6. **Simplifying the Binomial Coefficients**:
We know:
\[ \binom{n}{5} = \frac{n!}{5!(n-5)!} \quad \text{and} \quad \binom{n}{4} = \frac{n!}{4!(n-4)!} \]
Therefore:
\[ \frac{\binom{n}{5}}{\binom{n}{4}} = \frac{4!(n-4)!}{5!(n-5)!} = \frac{1}{5} \cdot \frac{(n-4)!}{(n-5)!} = \frac{1}{5} \cdot (n-4) \]

7. **Final Result**:
Thus, we find:
\[ \frac{x}{y} = \frac{n-4}{5} \]

### Conclusion:
The value of \(\frac{x}{y}\) is \(\frac{n-4}{5}\).