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Find the sum of the infinite geometric series.\newline−6−92−278−8132+-6 - \frac{9}{2} - \frac{27}{8} - \frac{81}{32} + \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−6−92−278−8132+-6 - \frac{9}{2} - \frac{27}{8} - \frac{81}{32} + \newlineWrite your answer as an integer or a fraction in simplest form.\newline______
  1. Identify Terms: Identify the first term aa and the common ratio rr of the geometric series.\newlineThe first term aa is −6-6.\newlineTo find the common ratio rr, we divide the second term by the first term.\newliner=−9/2−6=34r = \frac{-9/2}{-6} = \frac{3}{4}
  2. Convergence Check: Determine if the series is convergent.\newlineA geometric series converges if the absolute value of the common ratio ∣r∣\lvert r \rvert is less than 11.\newlineIn this case, ∣34∣=0.75\lvert \frac{3}{4} \rvert = 0.75, which is less than 11.\newlineTherefore, the series is convergent.
  3. Infinite Series Formula: Use the formula for the sum of an infinite geometric series.\newlineThe sum SS of an infinite geometric series is given by the formula S=a1−rS = \frac{a}{1 - r}, where aa is the first term and rr is the common ratio.
  4. Calculate Sum: Substitute the values of aa and rr into the formula to find the sum.\newlineS=−61−(34)S = \frac{-6}{1 - \left(\frac{3}{4}\right)}\newlineS=−614S = \frac{-6}{\frac{1}{4}}\newlineS=(−6)×(41)S = (-6) \times \left(\frac{4}{1}\right)\newlineS=−24S = -24

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