1. The problem is to find the sum of the infinite geometric series $$12 - 6 + 3 - \ldots$$ if it converges. 2. The formula for the sum of an infinite geometric series with first term $a$ and common ratio $r$ (where $|r| < 1$) is: $$S = \frac{a}{1 - r}$$ 3. Identify the first term $a$ and the common ratio $r$: - First term $a = 12$ - Common ratio $r = \frac{-6}{12} = -\frac{1}{2}$ 4. Check if the series converges by verifying $|r| < 1$: $$\left| -\frac{1}{2} \right| = \frac{1}{2} < 1$$ So, the series converges. 5. Calculate the sum using the formula: $$S = \frac{12}{1 - (-\frac{1}{2})} = \frac{12}{1 + \frac{1}{2}} = \frac{12}{\frac{3}{2}} = 12 \times \frac{2}{3} = 8$$ 6. Therefore, the sum of the series is $8$. Final answer: 8