1. **Problem Statement:** Find the sum of the infinite geometric sequence with first term 20 and common ratio $\frac{1}{2}$. Determine if the sum exists and calculate it if possible. 2. **Formula for the sum of an infinite geometric series:** $$ S = \frac{a}{1 - r} $$ where $a$ is the first term and $r$ is the common ratio. 3. **Condition for convergence:** The infinite sum exists only if $|r| < 1$. 4. **Check the ratio:** Here, $r = \frac{1}{2}$, and $|\frac{1}{2}| = 0.5 < 1$, so the sum converges. 5. **Calculate the sum:** $$ S = \frac{20}{1 - \frac{1}{2}} = \frac{20}{\frac{1}{2}} $$ 6. **Simplify the denominator:** $$ \frac{20}{\cancel{\frac{1}{2}}} = 20 \times 2 = 40 $$ 7. **Final answer:** The sum of the infinite sequence is $40$. --- **Summary:** The infinite geometric series with first term 20 and ratio $\frac{1}{2}$ converges, and its sum is $40$.