Chapter SH, Problem 10.12EP

To determine

To find: The sum of the given infinite series if it exists.

Answer to Problem 10.12EP

The sum of the given infinite series is 80.

Explanation of Solution

Given:

The given sequence is $20+15+11.25+\mathrm{...}$ .

Calculation:

The formula for the sum of an infinite geometric series with $-1<r<1$ is given by $S=\frac{a_1}{1-r}$ .

Sun will exist if the common ratio is between $-1<r<1$ .

Find the common ratio as follows.

  $\begin{array}{c}\frac{15}{20}=\frac{3}{4}\\ \frac{11.25}{15}=\frac{3}{4}\\ ⋮\end{array}$

The ratio is $\frac{3}{4}$ , whose absolute value is less than 1. So, some will exist.

Substitute 20 for $a_1$ and $\frac{3}{4}$ for $r$ in the equation $S=\frac{a_1}{1-r}$ to find the sum.

  $\begin{array}{c}S=\frac{20}{1-\frac{3}{4}}\\ =\frac{20}{\frac{1}{4}}\\ =80\end{array}$

Therefore, the sum of the given infinite series is 80.