1. **State the problem:** We want to express $$H_n = \frac{x^{\frac{n}{2}} - y^{\frac{n}{2}}}{x^{\frac{1}{2}} - y^{\frac{1}{2}}}$$ as a sum using the identity $$a^n - b^n = (a - b) \sum_{k=0}^{n-1} a^{n-1-k} b^k$$ where $n \geq 2$ is an integer. 2. **Recall the identity:** For any $a,b$ and integer $n$, $$a^n - b^n = (a - b) \sum_{k=0}^{n-1} a^{n-1-k} b^k$$ This means the difference of powers can be factored into a product of $(a-b)$ and a sum of terms. 3. **Apply the identity:** Let $$a = x^{\frac{1}{2}}, \quad b = y^{\frac{1}{2}}$$ Then, $$a^n - b^n = (x^{\frac{1}{2}})^n - (y^{\frac{1}{2}})^n = x^{\frac{n}{2}} - y^{\frac{n}{2}}$$ 4. **Rewrite $H_n$ using the identity:** $$H_n = \frac{x^{\frac{n}{2}} - y^{\frac{n}{2}}}{x^{\frac{1}{2}} - y^{\frac{1}{2}}} = \frac{a^n - b^n}{a - b} = \sum_{k=0}^{n-1} a^{n-1-k} b^k$$ 5. **Substitute back $a$ and $b$:** $$H_n = \sum_{k=0}^{n-1} \left(x^{\frac{1}{2}}\right)^{n-1-k} \left(y^{\frac{1}{2}}\right)^k = \sum_{k=0}^{n-1} x^{\frac{n-1-k}{2}} y^{\frac{k}{2}}$$ **Final answer:** $$\boxed{H_n = \sum_{k=0}^{n-1} x^{\frac{n-1-k}{2}} y^{\frac{k}{2}}}$$ This expresses $H_n$ as the sum of terms with powers of $x$ and $y$ increasing and decreasing by halves respectively.