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Find the sum of the infinite geometric series.\newline−1−12−14−18+-1 - \frac{1}{2} - \frac{1}{4} - \frac{1}{8} + \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−1−12−14−18+-1 - \frac{1}{2} - \frac{1}{4} - \frac{1}{8} + \newlineWrite your answer as an integer or a fraction in simplest form.\newline__\_\_
  1. Identify first term and common 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 first term is the first number in the series, and the common ratio is the factor by which we multiply to get from one term to the next.\newlineIn this series, the first term a=−1a = -1, and the common ratio r=12r = \frac{1}{2} (because each term is half of the previous term).
  2. Apply sum formula: The sum SS of an infinite geometric series with ∣r∣<1|r| < 1 can be found using the formula S=a1−rS = \frac{a}{1 - r}, where aa is the first term and rr is the common ratio.\newlineLet's apply this formula to our series.
  3. Substitute values into formula: Substitute the values of aa and rr into the formula: S=−11−12S = -\frac{1}{1 - \frac{1}{2}}
  4. Calculate denominator: Now, calculate the denominator:\newline1−12=121 - \frac{1}{2} = \frac{1}{2}
  5. Calculate sum: Next, calculate the sum using the values:\newlineS=−1(12)S = -\frac{1}{\left(\frac{1}{2}\right)}
  6. Multiply by reciprocal: To divide by a fraction, we multiply by its reciprocal. So, we multiply −1-1 by 22 (the reciprocal of 12\frac{1}{2}):\newlineS=−1×2S = -1 \times 2
  7. Perform final multiplication: Perform the multiplication to find the sum: \newlineS=−2S = -2

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