Question

In Problems 39– 44, find each sum.

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

Short answer

The sum of the given sequence is$2·1-{\frac{2}{3}}^n$

Step-by-step solution

Step 1. Write the given information.

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

Step 2. Find the common ratio.

$$\begin{gathered} a_1=\frac{2}{3},a_2={\frac{2}{3}}^2,a_3={\frac{2}{3}}^3 \\ \frac{a_2}{a_1}=\frac{{{\displaystyle \frac{2}{3}}}^2}{{\displaystyle \frac{2}{3}}} \\ =\frac{2}{3} \\ \frac{a_3}{a_2}=\frac{{{\displaystyle \frac{2}{3}}}^3}{{{\displaystyle \frac{2}{3}}}^2} \\ =\frac{2}{3} \\ r=\frac{2}{3} \end{gathered}$$

Step 3. Find the sum.

The formula for the sum is

$$\begin{gathered} s_n=\frac{a_1(1-r^n)}{1-r} \\ {s}_n=\frac{2}{3}·\frac{1-{{\displaystyle \frac{2}{3}}}^n}{1-{\displaystyle \frac{2}{3}}} \\ =\frac{2}{3}·\frac{1-{{\displaystyle \frac{2}{3}}}^n}{{\displaystyle \frac{1}{3}}} \\ =2·1-{\frac{2}{3}}^n \end{gathered}$$