Question

In Problems 39– 44, find each sum.

$2+\frac{6}{5}+\frac{18}{25}+...+2{\left(\frac{3}{5}\right)}^{n-1}$

Short answer

The sum of the given sequence is$5·1-{\frac{3}{5}}^n$

Step-by-step solution

Step 1. Write the information.

The given sequence is$2+\frac{6}{5}+\frac{18}{25}+...+2{\left(\frac{3}{5}\right)}^{n-1}$

Step 2. Find the common ratio.

$$\begin{gathered} a_1=2,a_2=\frac{6}{5},a_3=\frac{18}{25} \\ \frac{a_2}{a_1}=\frac{{\displaystyle \frac{6}{5}}}{2} \\ =\frac{3}{5} \\ \frac{a_3}{a_2}=\frac{{\displaystyle \frac{18}{25}}}{{\displaystyle \frac{6}{5}}} \\ =\frac{3}{5} \\ r=\frac{3}{5} \end{gathered}$$

Step 3. Find the sum

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

$$\begin{gathered} s_n=2·\frac{1-{{\displaystyle \frac{3}{5}}}^n}{1-{\displaystyle \frac{3}{5}}} \\ =2·\frac{1-{{\displaystyle \frac{3}{5}}}^n}{{\displaystyle \frac{2}{5}}} \\ =5·1-{\frac{3}{5}}^n \end{gathered}$$