Sum Fractions Step C47E4E

1. **State the problem:** We want to find the sum of the series $$\frac{1}{50}+\frac{1}{100}+\cdots+\frac{1}{505}+\frac{1}{510}$$ where the denominators increase by 5 each time starting from 50 up to 510. 2. **Identify the terms:** The terms are $$\frac{1}{50}, \frac{1}{55}, \frac{1}{60}, \ldots, \frac{1}{505}, \frac{1}{510}$$ because the denominators increase by 5. 3. **Number of terms:** To find the number of terms, use the formula for an arithmetic sequence: $$n = \frac{\text{last term} - \text{first term}}{\text{step}} + 1 = \frac{510 - 50}{5} + 1 = \frac{460}{5} + 1 = 92 + 1 = 93$$ 4. **Sum of the series:** The sum is $$S = \sum_{k=0}^{92} \frac{1}{50 + 5k}$$ 5. **No simple closed form:** This sum does not simplify nicely to elementary functions, so we approximate by calculating each term or using numerical methods. 6. **Approximate sum:** Calculate each term and add or use a numerical tool to find the approximate value. **Final answer:** The sum is $$\sum_{k=0}^{92} \frac{1}{50 + 5k}$$ which can be approximated numerically.