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Find the sum of the infinite geometric series.\newline8+6+92+278+…8 + 6 + \frac{9}{2} + \frac{27}{8} + \dots\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.\newline8+6+92+278+…8 + 6 + \frac{9}{2} + \frac{27}{8} + \dots\newlineWrite your answer as an integer or a fraction in simplest form.\newline__\_\_
  1. Identify Terms and Ratio: First, we need to identify the first term aa and the common ratio rr of the geometric series. The first term aa is the first number in the series, which is 88. To find the common ratio rr, we divide the second term by the first term, the third term by the second term, and so on, to ensure they all give the same ratio. r=68=92/6=278/92r = \frac{6}{8} = \frac{9}{2} / 6 = \frac{27}{8} / \frac{9}{2} r=34r = \frac{3}{4}
  2. Calculate Common Ratio: Now that we have the first term a=8a = 8 and the common ratio r=34r = \frac{3}{4}, which is less than 11, we can use the formula for the sum of an infinite geometric series:\newlineS=a(1−r)S = \frac{a}{(1 - r)}
  3. Use Formula for Sum: Let's plug the values into the formula to find the sum:\newlineS=81−34S = \frac{8}{1 - \frac{3}{4}}\newlineS=814S = \frac{8}{\frac{1}{4}}\newlineS=8×41S = 8 \times \frac{4}{1}\newlineS=32S = 32
  4. Plug in Values: We have calculated the sum of the series to be 3232. This is the final answer, and we should check it once more to ensure there are no errors.

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