1. **State the problem:** We need to evaluate the sum $$\sum_{x=2}^5 (x - 2x + 1)$$. 2. **Simplify the expression inside the sum:** $$x - 2x + 1 = (1x - 2x) + 1 = -x + 1$$ 3. **Rewrite the sum with the simplified expression:** $$\sum_{x=2}^5 (-x + 1)$$ 4. **Calculate each term individually:** - For $x=2$: $-(2) + 1 = -2 + 1 = -1$ - For $x=3$: $-(3) + 1 = -3 + 1 = -2$ - For $x=4$: $-(4) + 1 = -4 + 1 = -3$ - For $x=5$: $-(5) + 1 = -5 + 1 = -4$ 5. **Sum all terms:** $$-1 + (-2) + (-3) + (-4) = -1 - 2 - 3 - 4 = -10$$ **Final answer:** $$\sum_{x=2}^5 (x - 2x + 1) = -10$$