1. **State the problem:** Calculate the sum $$\sum_{x=1}^6 (-1)^x 3^x$$. 2. **Formula and explanation:** This is a finite sum of terms where each term is $$(-1)^x 3^x$$. We can write each term as $$(-3)^x$$ because $$(-1)^x 3^x = (-3)^x$$. 3. **Rewrite the sum:** $$\sum_{x=1}^6 (-3)^x = (-3)^1 + (-3)^2 + (-3)^3 + (-3)^4 + (-3)^5 + (-3)^6$$ 4. **Recognize this as a geometric series:** The sum of a geometric series with first term $$a$$ and common ratio $$r$$ for $$n$$ terms is: $$S_n = a \frac{1-r^n}{1-r}$$ Here, $$a = -3$$, $$r = -3$$, and $$n = 6$$. 5. **Calculate the sum:** $$S_6 = -3 \times \frac{1 - (-3)^6}{1 - (-3)}$$ 6. **Calculate powers and simplify:** $$(-3)^6 = 729$$ So, $$S_6 = -3 \times \frac{1 - 729}{1 + 3} = -3 \times \frac{-728}{4}$$ 7. **Simplify the fraction:** $$-3 \times \frac{\cancel{-728}}{\cancel{4}} = -3 \times (-182)$$ 8. **Multiply:** $$-3 \times (-182) = 546$$ **Final answer:** $$\sum_{x=1}^6 (-1)^x 3^x = 546$$