1. **State the problem:** We have a function defined by the functional equation $$f(x+1) = x - f(x)$$ with the initial condition $$f(1) = 1$$. We need to find $$f(2025)$$. 2. **Understand the functional equation:** The equation relates the value of the function at $$x+1$$ to the value at $$x$$. We can use this to find a pattern by computing the first few values. 3. **Calculate initial values:** - For $$x=1$$, $$f(1) = 1$$ (given). - For $$x=1$$, $$f(2) = 1 - f(1) = 1 - 1 = 0$$. - For $$x=2$$, $$f(3) = 2 - f(2) = 2 - 0 = 2$$. - For $$x=3$$, $$f(4) = 3 - f(3) = 3 - 2 = 1$$. - For $$x=4$$, $$f(5) = 4 - f(4) = 4 - 1 = 3$$. 4. **List the values:** $$f(1) = 1, f(2) = 0, f(3) = 2, f(4) = 1, f(5) = 3$$. 5. **Look for a pattern:** Let's check if the function repeats or follows a cycle. 6. **Calculate $$f(6)$$:** $$f(6) = 5 - f(5) = 5 - 3 = 2$$. 7. **Calculate $$f(7)$$:** $$f(7) = 6 - f(6) = 6 - 2 = 4$$. 8. **Calculate $$f(8)$$:** $$f(8) = 7 - f(7) = 7 - 4 = 3$$. 9. **Calculate $$f(9)$$:** $$f(9) = 8 - f(8) = 8 - 3 = 5$$. 10. **Calculate $$f(10)$$:** $$f(10) = 9 - f(9) = 9 - 5 = 4$$. 11. **Observe the pattern:** The values are: $$1, 0, 2, 1, 3, 2, 4, 3, 5, 4, ...$$ 12. **Try to find a formula:** Let's try to find a relation for even and odd values. 13. **Check odd values:** - $$f(1) = 1$$ - $$f(3) = 2$$ - $$f(5) = 3$$ - $$f(7) = 4$$ - $$f(9) = 5$$ This suggests for odd $$n=2k+1$$, $$f(n) = k+1$$. 14. **Check even values:** - $$f(2) = 0$$ - $$f(4) = 1$$ - $$f(6) = 2$$ - $$f(8) = 3$$ - $$f(10) = 4$$ This suggests for even $$n=2k$$, $$f(n) = k-1$$. 15. **Verify the formula:** - For odd $$n=2k+1$$, $$f(n) = k+1$$. - For even $$n=2k$$, $$f(n) = k-1$$. 16. **Apply to $$n=2025$$:** Since 2025 is odd, write $$2025 = 2k + 1$$. Solve for $$k$$: $$2025 = 2k + 1 \implies 2k = 2024 \implies k = 1012$$. Then, $$f(2025) = k + 1 = 1012 + 1 = 1013$$. **Final answer:** $$\boxed{1013}$$