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Find the sum of the infinite geometric series.\newline−7−73−79−727+-7 - \frac{7}{3} - \frac{7}{9} - \frac{7}{27} + \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.\newline−7−73−79−727+-7 - \frac{7}{3} - \frac{7}{9} - \frac{7}{27} + \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. In this series, the first term aa is −7-7, and we can find the common ratio by dividing the second term by the first term, or the third term by the second term, and so on. Let's calculate the common ratio rr: r=−7/3−7=13r = \frac{-7/3}{-7} = \frac{1}{3}.
  2. Calculate Common Ratio: Now that we have the first term a=−7a = -7 and the common ratio r=13r = \frac{1}{3}, we can use the formula for the sum of an infinite geometric series: S=a1−rS = \frac{a}{1 - r}. Let's plug in the values: S=−71−13S = \frac{-7}{1 - \frac{1}{3}}.
  3. Apply Formula: We need to simplify the expression: S=(−7)/(1−13)=(−7)/(23)S = (-7) / (1 - \frac{1}{3}) = (-7) / (\frac{2}{3}).\newlineTo divide by a fraction, we multiply by its reciprocal: S=(−7)×(32)S = (-7) \times (\frac{3}{2}).
  4. Simplify Expression: Now, let's perform the multiplication: S=(−7)×(32)=−212S = (-7) \times \left(\frac{3}{2}\right) = -\frac{21}{2}. This is the sum of the infinite geometric series in its simplest form.

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