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Find the sum of the infinite geometric series.\newline5+154+4516+13564+…5 + \frac{15}{4} + \frac{45}{16} + \frac{135}{64} + \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.\newline5+154+4516+13564+…5 + \frac{15}{4} + \frac{45}{16} + \frac{135}{64} + \dots\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 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 55, and we can find the common ratio by dividing the second term by the first term.
  2. Calculate common ratio: Calculate the common ratio rr by dividing the second term 154\frac{15}{4} by the first term 55.r=1545r = \frac{\frac{15}{4}}{5}r=154×15r = \frac{15}{4} \times \frac{1}{5}r=1520r = \frac{15}{20}r=34r = \frac{3}{4}
  3. Use formula for sum: Now that we have the first term a=5a = 5 and the common ratio r=34r = \frac{3}{4}, we can use the formula for the sum of an infinite geometric series.S=a1−rS = \frac{a}{1 - r}S=51−34S = \frac{5}{1 - \frac{3}{4}}
  4. Calculate sum: Calculate the sum SS by evaluating the expression.\newlineS=51−34S = \frac{5}{1 - \frac{3}{4}}\newlineS=544−34S = \frac{5}{\frac{4}{4} - \frac{3}{4}}\newlineS=514S = \frac{5}{\frac{1}{4}}\newlineS=5×41S = 5 \times \frac{4}{1}\newlineS=20S = 20

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