1. **Problem statement:** Find a formula for the sum $$\sum_{i=1}^x i$$ assuming $x$ is even. 2. **Recall the problem:** We want to find a closed-form expression for the sum of the first $x$ natural numbers. 3. **Formula used:** The sum of the first $x$ natural numbers is given by the arithmetic series formula: $$\sum_{i=1}^x i = \frac{x}{2} (\text{first term} + \text{last term})$$ Since the first term is 1 and the last term is $x$, this becomes: $$\sum_{i=1}^x i = \frac{x}{2} (1 + x)$$ 4. **Explanation:** - The sum of an arithmetic series is the average of the first and last terms multiplied by the number of terms. - Here, the number of terms is $x$. - Since $x$ is even, $\frac{x}{2}$ is an integer, making the formula straightforward. 5. **Intermediate step:** $$\sum_{i=1}^x i = \cancel{\frac{x}{2}} (1 + x)$$ 6. **Final formula:** $$\sum_{i=1}^x i = \frac{x(x+1)}{2}$$ This formula works for all positive integers $x$, including odd values, but the problem only requires the even case. **Answer:** $$\boxed{\sum_{i=1}^x i = \frac{x(x+1)}{2}}$$