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Find the sum of the infinite geometric series.\newline−6−2−23−29+-6 - 2 - \frac{2}{3} - \frac{2}{9} + \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−6−2−23−29+-6 - 2 - \frac{2}{3} - \frac{2}{9} + \newlineWrite your answer as an integer or a fraction in simplest form.\newline______
  1. Identify terms and ratio: Identify the first term a1a_1 and the common ratio rr of the geometric series.a1=−6a_1 = -6To find rr, we use the formula r=a2a1r = \frac{a_2}{a_1}.r=−2−6=13r = \frac{-2}{-6} = \frac{1}{3}
  2. Apply sum formula: Apply the formula for the sum of an infinite geometric series, which is S=a11−rS = \frac{a_1}{1 - r}, where SS is the sum, a1a_1 is the first term, and rr is the common ratio.\newlineHere, a1=−6a_1 = -6 and r=13r = \frac{1}{3}.\newlineCalculate the sum using these values.\newlineS=−61−(13)S = \frac{-6}{1 - (\frac{1}{3})}
  3. Simplify expression: Simplify the expression to find the sum.\newlineS=−623S = \frac{-6}{\frac{2}{3}}\newlineTo divide by a fraction, multiply by its reciprocal.\newlineS=(−6)×32S = (-6) \times \frac{3}{2}
  4. Perform multiplication: Perform the multiplication to get the final answer.\newlineS=−182S = \frac{-18}{2}\newlineS=−9S = -9

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