Chapter 11.6, Problem 3WE

To determine

To calculate:

Sum of given geometric series.

Answer to Problem 3WE

Sum of given geometric series is $9.6$ .

Explanation of Solution

Give information:

  $27-18+12-8+\mathrm{...}$

Calculation:

In order to find out common ratio of given number,

  $27-18+12-8+\mathrm{...}$

Here first term is $a_1=27$

So common ratio is $r=\frac{-18}{27}=\frac{-2}{3}$

Since $\left|r\right|=\left|\frac{-2}{3}\right|<1$

Infinite sum exists.

So, Sum of given geometric series is calculated as,

  $=\frac{a^n}{1-r}$

Where,

  $a_1=$ Initial term.

  $r=$ common ratio between terms.

  $n=$ number of terms

Put the values in the equation.

  $\begin{array}{l}=\frac{a^n}{1-r}\\ =\frac{27}{1-\left(-\frac{2}{3}\right)}\\ =\frac{24}{1+\frac{2}{3}}\\ =\frac{24}{\frac{5}{2}}\\ =\frac{2\left(24\right)}{5}\\ =\frac{48}{5}\\ =9.6\end{array}$

Therefore, sum of given geometric series is $9.6$ .