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Find the sum of the infinite geometric series.\newline−4−3−94−2716+-4 - 3 - \frac{9}{4} - \frac{27}{16} + \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−4−3−94−2716+-4 - 3 - \frac{9}{4} - \frac{27}{16} + \newlineWrite your answer as an integer or a fraction in simplest form.\newline______
  1. Identify Terms and 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 formula for the sum of an infinite geometric series is S=a1−rS = \frac{a}{1 - r}, where ∣r∣<1|r| < 1. In the given series, the first term is −4-4, and we can find the common ratio by dividing the second term by the first term. Common ratio rr = second term / first term = −3/−4=34-3 / -4 = \frac{3}{4}.
  2. Calculate Common Ratio: Now that we have the first term a=−4a = -4 and the common ratio r=34r = \frac{3}{4}, we can use the formula for the sum of an infinite geometric series to find the sum.S=a1−r=−41−34S = \frac{a}{1 - r} = \frac{-4}{1 - \frac{3}{4}}.
  3. Use Formula for Sum: We simplify the denominator of the fraction:\newline1−34=44−34=141 - \frac{3}{4} = \frac{4}{4} - \frac{3}{4} = \frac{1}{4}.
  4. Simplify Denominator: Now we substitute the simplified denominator back into the formula:\newlineS=−4(14)S = -\frac{4}{\left(\frac{1}{4}\right)}.
  5. Substitute and Simplify: To divide by a fraction, we multiply by its reciprocal. So, we multiply −4-4 by the reciprocal of 14\frac{1}{4}, which is 41\frac{4}{1}.\newlineS=−4×(41)=−16S = -4 \times \left(\frac{4}{1}\right) = -16.
  6. Final Answer: The sum of the infinite geometric series is −16-16. This is the final answer.

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