Chapter 11.6, Problem 8OE

To determine

To calculate:

Common ratio of given geometric series and sum series has.

Answer to Problem 8OE

Common ratio of given geometric series is $1.1$ and sum is $2$ .

Explanation of Solution

Give information:

  ${\displaystyle \sum _{n=1}^{\infty }2{\left(1.1\right)}^n}$

Calculation:

In order to find out common ratio of given number,

  ${\displaystyle \sum _{n=1}^{\infty }2{\left(1.1\right)}^n}$

Common ratio $\left(r\right)$ is,

  $r=1.1$

Sum of given geometric series is calculated as,

  ${\displaystyle \sum _{k=0}^{n-1}\left(a{r}^k\right)}=a\frac{1-r^n}{1-r}$

Where,

  $a_1=$ Initial term.

  $r=$ common ratio between terms.

  $n=$ number of terms

Put the values in the equation.

  $\begin{array}{l}{\displaystyle \sum _{k=0}^{n-1}\left(a{r}^k\right)}\\ =a\frac{1-r^n}{1-r}\\ =2\frac{1-{\left(1.1\right)}^1}{1-\left(1.1\right)}\\ =2\end{array}$

Therefore, Common ratio of given geometric series is $1.1$ and sum is $2$ .