1. **Problem Statement:** Find $n$ such that $$\frac{x^{n-5}+y^{n-5}}{x^{n-6}+y^{n-6}}$$ is the Harmonic Mean (H.M.) between $x$ and $y$. 2. **Recall the Harmonic Mean formula:** $$H.M. = \frac{2}{\frac{1}{x} + \frac{1}{y}} = \frac{2xy}{x+y}$$ 3. **Set the given expression equal to the H.M.:** $$\frac{x^{n-5}+y^{n-5}}{x^{n-6}+y^{n-6}} = \frac{2xy}{x+y}$$ 4. **Analyze the expression by factoring powers:** Rewrite numerator and denominator: $$x^{n-6} \cdot x + y^{n-6} \cdot y = x^{n-5} + y^{n-5}$$ So, $$\frac{x^{n-5}+y^{n-5}}{x^{n-6}+y^{n-6}} = \frac{x \cdot x^{n-6} + y \cdot y^{n-6}}{x^{n-6} + y^{n-6}}$$ 5. **Consider the case $x = y$:** If $x = y$, then numerator and denominator become: $$\frac{2x^{n-5}}{2x^{n-6}} = x$$ The harmonic mean when $x = y$ is $x$, so this holds for any $n$. 6. **Check for $n=5$ explicitly:** Substitute $n=5$: $$\frac{x^{0} + y^{0}}{x^{-1} + y^{-1}} = \frac{1 + 1}{\frac{1}{x} + \frac{1}{y}} = \frac{2}{\frac{x+y}{xy}} = \frac{2xy}{x+y}$$ This matches the harmonic mean exactly. 7. **Conclusion:** The value of $n$ that makes the expression the harmonic mean is: $$n = 5$$