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Find the sum of the infinite geometric series.\newline1+35+925+27125+…1 + \frac{3}{5} + \frac{9}{25} + \frac{27}{125} + \dots\newlineWrite your answer as an integer or a fraction in simplest form.\newline__\_\_

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Q. Find the sum of the infinite geometric series.\newline1+35+925+27125+…1 + \frac{3}{5} + \frac{9}{25} + \frac{27}{125} + \dots\newlineWrite your answer as an integer or a fraction in simplest form.\newline__\_\_
  1. Identify Terms and Ratio: To find the sum of an infinite geometric series, we need to identify the first term aa and the common ratio rr of the series. The formula for the sum of an infinite geometric series is S=a1−rS = \frac{a}{1 - r}, where ∣r∣<1|r| < 1.
  2. Calculate Common Ratio: The first term aa of the series is 11. The second term is 35\frac{3}{5}, so to find the common ratio rr, we divide the second term by the first term: r=351=35r = \frac{\frac{3}{5}}{1} = \frac{3}{5}.
  3. Check Ratio Value: Now we check if the common ratio rr is less than 11 in absolute value. Since 35\frac{3}{5} is less than 11, we can use the sum formula for an infinite geometric series.
  4. Apply Sum Formula: We apply the formula S=a1−rS = \frac{a}{1 - r} to find the sum of the series. Substituting the values we have: S=11−35S = \frac{1}{1 - \frac{3}{5}}.
  5. Calculate Sum: We calculate the sum: S=11−35=125=1×52=52.S = \frac{1}{1 - \frac{3}{5}} = \frac{1}{\frac{2}{5}} = 1 \times \frac{5}{2} = \frac{5}{2}.
  6. Final Answer: The sum of the infinite geometric series is 52\frac{5}{2}. This is the final answer in its simplest form.

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