Chapter 11.6, Problem 9OE

To determine

To calculate:

Common ratio of given geometric series and sum series has.

Answer to Problem 9OE

Common ratio of given geometric series is $-0.35$ and sum is $1$ .

Explanation of Solution

Give information:

  ${\displaystyle \sum _{n=1}^{\infty }{\left(-0.35\right)}^n}$

Calculation:

In order to find out common ratio of given number,

  ${\displaystyle \sum _{n=1}^{\infty }{\left(-0.35\right)}^n}$

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

  $r=-0.35$

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.

  $\begin{array}{l}{\displaystyle \sum _{k=0}^{n-1}\left(a{r}^k\right)}\\ =a\frac{1-r^n}{1-r}\\ =1\frac{1-{\left(-0.35\right)}^1}{1-\left(-0.35\right)}\\ =1\end{array}$

Therefore, Common ratio of given geometric series is $-0.35$ and sum is $1$ .