1. **State the problem:** We have the infinite series $$\sum_{n=1}^\infty 2 \left(-\frac{1}{2}\right)^{n-1}$$ and we want to find the first term $a_1$, the common ratio $r$, and the sum of the series. 2. **Identify the first term $a_1$:** The first term is when $n=1$: $$a_1 = 2 \left(-\frac{1}{2}\right)^{1-1} = 2 \left(-\frac{1}{2}\right)^0 = 2 \times 1 = 2$$ 3. **Identify the common ratio $r$:** The common ratio is the factor multiplied to get from one term to the next. From the formula, it is: $$r = -\frac{1}{2}$$ 4. **Sum of an infinite geometric series:** The sum $S$ of an infinite geometric series with first term $a_1$ and common ratio $r$ (where $|r| < 1$) is given by: $$S = \frac{a_1}{1 - r}$$ 5. **Calculate the sum:** Substitute $a_1 = 2$ and $r = -\frac{1}{2}$: $$S = \frac{2}{1 - \left(-\frac{1}{2}\right)} = \frac{2}{1 + \frac{1}{2}} = \frac{2}{\frac{3}{2}}$$ 6. **Simplify the fraction:** $$S = 2 \times \frac{2}{3} = \frac{4}{3}$$ **Final answers:** - First term $a_1 = 2$ - Common ratio $r = -\frac{1}{2}$ - Sum of the series $S = \frac{4}{3}$