Chapter 8, Problem 20CR

To determine

Find the sum of the given geometric series.

Answer to Problem 20CR

The sum of the given series is $-1.6$ .

Explanation of Solution

Given:

The given geometric seriesis $-2+\frac{1}{2}-\frac{1}{8}+\frac{1}{32}+---$ .

Calculation:

In angeometric sequence, the ratio of any term to the previous term, called the common ratio, is constant.

Find the common ratioof any term to the previous term of the sequence $-2+\frac{1}{2}-\frac{1}{8}+\frac{1}{32}+---$ .

  $\begin{array}{l}\frac{\frac{1}{2}}{-2}=-\frac{1}{4}\\ \frac{-\frac{1}{8}}{\frac{1}{2}}=-\frac{1}{4}\\ \frac{\frac{1}{32}}{-\frac{1}{8}}=-\frac{1}{4}\end{array}$

The common ratio $\left(-\frac{1}{4}\right)$ is constant.

The sum of the infinite geometric series is $s_n=\frac{a_1}{1-r}$ , where $a_1$ is the first number and $r$ is the common ratio.

In the given series $a_1=-4$ and $r=-\frac{1}{4}$ .

  $\begin{array}{l}{s}_n=\frac{-2}{1-\left(-\frac{1}{4}\right)}\\ =\frac{-2}{1+\frac{1}{4}}\\ =\frac{-2}{\frac{4+1}{4}}\\ =\frac{-2\cdot 4}{5}\\ =-\frac{8}{5}\\ =-1.6\end{array}$

Hence the sum of the given series is $-1.6$ .