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Find the sum of the infinite geometric series. $\newline$ $6 + \frac{9}{2} + \frac{27}{8} + \frac{81}{32} + \dots$ $\newline$ Write your answer as an integer or a fraction in simplest form. $\newline$ $\_\_$

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Find the value of an infinite geometric series

Full solution

Q. Find the sum of the infinite geometric series.

\newline

6 + \frac{9}{2} + \frac{27}{8} + \frac{81}{32} + \dots

\newline

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

\newline

\_\_

Identify first term and common ratio: First, we need to identify the first term $a$ and the common ratio $r$ of the geometric series. $\newline$ The first term $a$ is the first number in the series, which is $6$ . $\newline$ The common ratio $r$ is found by dividing the second term by the first term, which is $(\frac{9}{2}) / 6 = \frac{3}{4}$ .
Check common ratio: Next, we check if the absolute value of the common ratio is less than $1$ , which is necessary for the sum of an infinite geometric series to exist. $\newline$ Since $|\frac{3}{4}| = 0.75$ , which is less than $1$ , the series converges and we can find the sum.
Use formula for sum: Now, we use the formula for the sum of an infinite geometric series, which is $S = \frac{a}{1 - r}$ , where $S$ is the sum, $a$ is the first term, and $r$ is the common ratio. $\newline$ Substitute the values of $a$ and $r$ into the formula: $S = \frac{6}{1 - \frac{3}{4}}$ .
Calculate the sum: We calculate the sum: $S = \frac{6}{1 - \frac{3}{4}} = \frac{6}{\frac{1}{4}} = 6 \times 4 = 24$ .

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