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Find the sum of the infinite geometric series.\newline−10−5−52−54+⋯-10 - 5 - \frac{5}{2} - \frac{5}{4} + \cdots\newline\newlineWrite your answer as an integer or a fraction in simplest form.\newline‾\underline{\hspace{3em}}\newline

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Q. Find the sum of the infinite geometric series.\newline−10−5−52−54+⋯-10 - 5 - \frac{5}{2} - \frac{5}{4} + \cdots\newline\newlineWrite your answer as an integer or a fraction in simplest form.\newline‾\underline{\hspace{3em}}\newline
  1. Identify the first term: Identify the first term a1a_1 of the geometric series.\newlineThe first term a1a_1 is the first number in the series, which is −10-10.
  2. Determine the common ratio: Determine the common ratio rr of the geometric series.\newlineThe common ratio rr can be found by dividing the second term by the first term: r=a2a1=−5−10=12r = \frac{a_2}{a_1} = \frac{-5}{-10} = \frac{1}{2}.
  3. Use the formula for the sum: Use the formula for the sum of an infinite geometric series.\newlineThe formula 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.
  4. Plug the values into the formula: Plug the values of a1a_1 and rr into the formula to find the sum.\newlineS=−101−(12)=−1012=(−10)⋅21=−20S = \frac{-10}{1 - (\frac{1}{2})} = \frac{-10}{\frac{1}{2}} = (-10) \cdot \frac{2}{1} = -20.

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