Infinite Series Sum 5F6F61

1. **State the problem:** We need to find the sum of the infinite geometric series $$5 + 2.5 + 1.25 + 0.625 + ...$$ 2. **Identify the first term and common ratio:** - The first term $$a = 5$$ - The common ratio $$r = \frac{2.5}{5} = 0.5$$ 3. **Recall the formula for the sum of an infinite geometric series:** $$S = \frac{a}{1 - r}$$ where $$|r| < 1$$. 4. **Check the condition:** Since $$|0.5| < 1$$, the formula applies. 5. **Calculate the sum:** $$S = \frac{5}{1 - 0.5} = \frac{5}{0.5}$$ 6. **Simplify the fraction:** $$S = \frac{5}{\cancel{0.5}} \times \frac{2}{2} = \frac{5 \times 2}{1} = 10$$ 7. **Final answer:** The sum of the infinite geometric series is $$10$$.