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Find the sum of the infinite geometric series.\newline2+43+89+1627+…2 + \frac{4}{3} + \frac{8}{9} + \frac{16}{27} + \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.\newline2+43+89+1627+…2 + \frac{4}{3} + \frac{8}{9} + \frac{16}{27} + \dots\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.
  2. Calculate Common Ratio: The first term aa of the series is 22. The second term is 43\frac{4}{3}, so to find the common ratio rr, we divide the second term by the first term: r=432=43×12=23r = \frac{\frac{4}{3}}{2} = \frac{4}{3} \times \frac{1}{2} = \frac{2}{3}.
  3. Apply Formula for Sum: Now that we have the first term and the common ratio, we can use the formula for the sum of an infinite geometric series: S=a1−rS = \frac{a}{1 - r}. Plugging in the values we have S=21−23S = \frac{2}{1 - \frac{2}{3}}.
  4. Simplify Expression: Simplify the expression: S=21−23=233−23=213=2×3=6S = \frac{2}{1 - \frac{2}{3}} = \frac{2}{\frac{3}{3} - \frac{2}{3}} = \frac{2}{\frac{1}{3}} = 2 \times 3 = 6.

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