Chapter 11.6, Problem 2OE

To determine

To calculate:

Common ratio of given geometric series and check whether the series has a sum.

Answer to Problem 2OE

Common ratio of given geometric series is $-\frac{1}{3}$ and a sum is $20.5$ .

Explanation of Solution

Give information:

  $27-9+\mathrm{3...}$

Calculation:

In order to find out common ratio of given number,

  $27-9+3$

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

  $\begin{array}{l}=\frac{t_2}{t_1}\\ =\frac{-9}{27}\\ =\frac{-1}{3}\end{array}$

Or

  $\begin{array}{l}=\frac{3}{-9}\\ =-\frac{1}{3}\end{array}$

Sum of given geometric series is calculated as,

  $S=\frac{a_1}{1-r}$

Where,

  $a_1=$ Initial term.

Put the values in the equation.

  $\begin{array}{l}S=\frac{27}{1-\left(-1/3\right)}\\ S=\frac{\left(3\right)\left(27\right)}{4}\\ S=\frac{81}{4}\\ S=20.5\end{array}$

Therefore, Common ratio of given geometric series is $-\frac{1}{3}$ and a sum is $20.5$ .