Question

For Problems 45–50, use a graphing utility to find the sum of each geometric sequence.

$\sum _{n=1}^{15}{\left(\frac{2}{3}\right)}^n$

Short answer

The sum of the given sequence is$1.9954$

Step-by-step solution

Step 1. Write the given information.

The given sequence is $\sum _{n=1}^{15}{\left(\frac{2}{3}\right)}^n$

Step 2. Find the common ratio.

$$\begin{gathered} a_1=\frac{2}{3},a_2={\frac{2}{3}}^2 \\ \frac{a_2}{a_1}=\frac{{{\displaystyle \frac{2}{3}}}^2}{{\displaystyle \frac{2}{3}}} \\ =\frac{2}{3} \\ r=\frac{2}{3} \end{gathered}$$

Step 3. Find the sum.

The formula for the sum of nth term is $s_n=\frac{a_1(1-r^n)}{1-r}$

$$\begin{gathered} s_{15}=\frac{2}{3}·\frac{1-{{\displaystyle \frac{2}{3}}}^{15}}{1-{\displaystyle \frac{2}{3}}} \\ =\frac{2}{3}·\frac{1-{{\displaystyle \frac{2}{3}}}^{15}}{{\displaystyle \frac{1}{3}}} \\ =2·1-{\frac{2}{3}}^{15} \\ =1.9954 \end{gathered}$$