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 series is $-2+\frac{1}{2}-\frac{1}{8}+\frac{1}{32}+---$ .

Calculation:

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

Find the common ratio of 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 $-1.6$ .