Chapter 11.5, Problem 18WE

To determine

To calculate:

Sum of given geometric series.

Answer to Problem 18WE

Sum of given geometric series is $\frac{-1}{3}$ .

Explanation of Solution

Give information:

  ${\displaystyle \sum _{j=1}^{10}{\left(-\frac{1}{2}\right)}^j}$

Calculation:

In order to find out sum,

  ${\displaystyle \sum _{j=1}^{10}{\left(-\frac{1}{2}\right)}^j}$

Sum of given geometric series is calculated as,

Here $a={\left(\frac{-1}{2}\right)}^1=-\frac{1}{2},r=-\frac{1}{2}$

Infinite geometric progression is calculated as,

  $S=\frac{a}{1-r}$

Where,

  $S=$ Sum of geometric progression.

  $a=$ First term,

  $r=$ common ratio.

Put the values in above equation,

  $\begin{array}{l}S=\frac{-\frac{1}{2}}{1-\left(-\frac{1}{2}\right)}\\ S=\frac{-\frac{1}{2}}{1+\frac{1}{2}}\\ S=\frac{-\frac{1}{2}}{\frac{3}{2}}\\ S=\frac{2\left(-\frac{1}{2}\right)}{3}\\ S=\frac{-1}{3}\end{array}$

Therefore, sum of given geometric series is $\frac{-1}{3}$ .