Find the sum of the infinite geometric series. 3 + 9/5 + 27/25 + 81/125 + 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 3. To find the common ratio (r), we divide the second term by the first term: (9/5) / 3 = 9/5 * 1/3 = 3/5. So, the common ratio (r) is 3/5.Apply Sum Formula: Now we can apply the formula for the sum of an infinite geometric series: S = a / (1 - r). Plugging in the values we have S = 3 / (1 - 3/5).Simplify Denominator: Simplify the denominator: 1 - 3/5 = 5/5 - 3/5 = 2/5.Calculate Final Sum: Now, calculate the sum: S = 3 / (2/5) = 3 * (5/2) = 15/2.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the sum of the infinite geometric series. $\newline$ $3 + \frac{9}{5} + \frac{27}{25} + \frac{81}{125} + \dots$ $\newline$ Write your answer as an integer or a fraction in simplest form. $\newline$ ______

View step-by-step help

Home

Math Problems

Precalculus

Find the value of an infinite geometric series

Full solution

Q. Find the sum of the infinite geometric series.

\newline

3 + \frac{9}{5} + \frac{27}{25} + \frac{81}{125} + \dots

\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 $3$ . To find the common ratio $r$ , we divide the second term by the first term: $\frac{9}{5} / 3 = \frac{9}{5} \times \frac{1}{3} = \frac{3}{5}$ . So, the common ratio $r$ is $\frac{3}{5}$ .
Apply Sum Formula: Now we can apply the formula for the sum of an infinite geometric series: $S = \frac{a}{1 - r}$ . Plugging in the values we have $S = \frac{3}{1 - \frac{3}{5}}$ .
Simplify Denominator: Simplify the denominator: $1 - \frac{3}{5} = \frac{5}{5} - \frac{3}{5} = \frac{2}{5}$ .
Calculate Final Sum: Now, calculate the sum: $S = \frac{3}{\left(\frac{2}{5}\right)} = 3 \times \left(\frac{5}{2}\right) = \frac{15}{2}$ .