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Find the sum of the infinite geometric series.\newline4+125+3625+108125+4 + \frac{12}{5} + \frac{36}{25} + \frac{108}{125} + \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.\newline4+125+3625+108125+4 + \frac{12}{5} + \frac{36}{25} + \frac{108}{125} + \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 the first number in the sequence, which is 44. The common ratio rr can be found by dividing the second term by the first term, the third term by the second term, and so on. Let's calculate rr using the first two terms: r=125/4r = \frac{12}{5} / 4.
  3. Use Formula for Sum: Calculating the common ratio: r=125/4=125⋅14=1220=35.r = \frac{12}{5} / 4 = \frac{12}{5} \cdot \frac{1}{4} = \frac{12}{20} = \frac{3}{5}.
  4. Plug in Values: Now that we have the first term a=4a = 4 and the common ratio r=35r = \frac{3}{5}, we can use the formula for the sum of an infinite geometric series: S=a1−rS = \frac{a}{1 - r}.
  5. Calculate Sum: Plugging the values into the formula: S=41−35=425=4×52=202=10S = \frac{4}{1 - \frac{3}{5}} = \frac{4}{\frac{2}{5}} = 4 \times \frac{5}{2} = \frac{20}{2} = 10.
  6. Calculate Sum: Plugging the values into the formula: S=41−35=425=4×52=202=10S = \frac{4}{1 - \frac{3}{5}} = \frac{4}{\frac{2}{5}} = 4 \times \frac{5}{2} = \frac{20}{2} = 10.The sum of the infinite geometric series is 1010. This is an integer and it is in its simplest form.

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