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Find the sum of the infinite geometric series.\newline−7−74−716−764+-7 - \frac{7}{4} - \frac{7}{16} - \frac{7}{64} + \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−7−74−716−764+-7 - \frac{7}{4} - \frac{7}{16} - \frac{7}{64} + \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 the given series, the first term is −7-7 and each subsequent term is obtained by multiplying the previous term by 14\frac{1}{4}. Therefore, the common ratio rr is 14\frac{1}{4}.
  2. Apply Sum Formula: We can now apply the formula for the sum of an infinite geometric series: S=a1−rS = \frac{a}{1 - r}. Substituting the values we have, S=−71−14S = \frac{-7}{1 - \frac{1}{4}}.
  3. Substitute Values: Now we perform the calculation: S=(−7)/(1−1/4)=(−7)/(3/4)=(−7)×(4/3)=−28/3S = (-7) / (1 - 1/4) = (-7) / (3/4) = (-7) \times (4/3) = -28/3.
  4. Perform Calculation: The sum of the infinite geometric series is −283-\frac{28}{3}. This is the final answer, and it is in its simplest form.

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