Find the sum of the infinite geometric series. 6 + 18/5 + 54/25 + 162/125 + Write your answer as an integer or a fraction in simplest form. ______

Identify Terms and Ratio: To find the sum of an infinite geometric series, we need to identify the first term (a) and the common ratio (r) of the series. The formula for the sum of an infinite geometric series is S = a / (1 - r), where |r| < 1.Calculate Common Ratio: The first term (a) of the series is the first number in the sequence, which is 6. The common ratio (r) 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 r using the first two terms: r = (18/5) / 6.Use Formula for Sum: Calculating the common ratio: r = (18/5) / 6 = (18/5) * (1/6) = 18 / 30 = 3 / 5.Plug in Values: Now that we have the first term (a = 6) and the common ratio (r = 3/5), we can use the formula for the sum of an infinite geometric series: S = a / (1 - r).Calculate Sum: Plugging the values into the formula: S = 6 / (1 - 3/5) = 6 / (2/5) = 6 * (5/2) = 30/2 = 15.Final Answer: The sum of the infinite geometric series is 15. This is the final answer.

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Find the sum of the infinite geometric series. $\newline$ $6 + \frac{18}{5} + \frac{54}{25} + \frac{162}{125} +$ $\newline$ Write your answer as an integer or a fraction in simplest form. $\newline$ $\_\_$

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Precalculus

Find the value of an infinite geometric series

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Q. Find the sum of the infinite geometric series.

\newline

6 + \frac{18}{5} + \frac{54}{25} + \frac{162}{125} +

\newline

Write your answer as an integer or a fraction in simplest form.

\newline

\_\_

Identify Terms and Ratio: To find the sum of an infinite geometric series, we need to identify the first term $a$ and the common ratio $r$ of the series. The formula for the sum of an infinite geometric series is $S = \frac{a}{1 - r}$ , where |r| < 1.
Calculate Common Ratio: The first term $a$ of the series is the first number in the sequence, which is $6$ . The common ratio $r$ 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 $r$ using the first two terms: $r = \frac{18}{5} / 6$ .
Use Formula for Sum: Calculating the common ratio: $r = \frac{18}{5} / 6 = \frac{18}{5} \cdot \frac{1}{6} = \frac{18}{30} = \frac{3}{5}$ .
Plug in Values: Now that we have the first term $a = 6$ and the common ratio $r = \frac{3}{5}$ , we can use the formula for the sum of an infinite geometric series: $S = \frac{a}{1 - r}$ .
Calculate Sum: Plugging the values into the formula: $S = \frac{6}{1 - \frac{3}{5}} = \frac{6}{\frac{2}{5}} = 6 \times \frac{5}{2} = \frac{30}{2} = 15$ .
Final Answer: The sum of the infinite geometric series is $15$ . This is the final answer.