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Find the sum of the infinite geometric series.\newline−9−274−8116−24364+-9 - \frac{27}{4} - \frac{81}{16} - \frac{243}{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−9−274−8116−24364+-9 - \frac{27}{4} - \frac{81}{16} - \frac{243}{64} + \newlineWrite your answer as an integer or a fraction in simplest form.\newline______
  1. Identify Terms: First, we need to identify the first term aa and the common ratio rr of the geometric series.\newlineThe first term is −9-9.\newlineTo find the common ratio, we divide the second term by the first term: (−27/4)/(−9)=3/4(-27/4) / (-9) = 3/4.\newlineSo, the common ratio rr is 3/43/4.
  2. Check Convergence: Next, we check if the absolute value of the common ratio is less than 11, which is a necessary condition for the convergence of an infinite geometric series.\newlineSince ∣34∣=0.75|\frac{3}{4}| = 0.75, which is less than 11, the series converges.
  3. Use Sum Formula: Now, we can use the formula for the sum of an infinite geometric series, which is S=a1−rS = \frac{a}{1 - r}, where SS is the sum, aa is the first term, and rr is the common ratio.\newlineLet's plug in the values: S=−91−(34)S = \frac{-9}{1 - (\frac{3}{4})}.
  4. Calculate Denominator: We calculate the denominator: 1−(34)=141 - \left(\frac{3}{4}\right) = \frac{1}{4}.
  5. Calculate Sum: Now, we calculate the sum: S=(−9)/(14)S = (-9) / (\frac{1}{4}).\newlineTo divide by a fraction, we multiply by its reciprocal: S=(−9)×(41)=−36S = (-9) \times (\frac{4}{1}) = -36.

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