Chapter 9, Problem 3PS

To determine

To show: The two series given in the table have finite sum and specify what the sums represent.

Answer to Problem 3PS

The sum of the first series is $40\text{ feet}$ and that of the second series is $2\text{ seconds}$ . The first sum represents the total distance travelled and the second sum denotes the time taken to cover the total distance.

Explanation of Solution

Given:

The table is:

Distance (in feet)	Time (in seconds)
20	1
10	0.5
5	0.25
2.5	0.125
1.25	0.0625
0.625	0.03125

Table 1

Calculation:

Let D denote the sum of the infinite distance series and T denote the sum of the infinite time series.

From Table 1, the first term, a , of the distance series is 20 and that of the time series is 1. The common ratio, r , for both the series is 0.5 which is clearly less than 1.

So, to determine the sum of both the series, the formula for sum to infinity of a geometric series can be applied here.

  $\begin{array}{c}D=\frac{a}{1-r}\\ =\frac{20}{1-0.5}\\ =40\text{ feet}\end{array}$

The sum implies that the total distance travelled is equal to $40\text{ feet}$ .

  $\begin{array}{c}T=\frac{a}{1-r}\\ =\frac{1}{1-0.5}\\ =2\text{ seconds}\end{array}$

So, the total time taken to travel the distance of $40\text{ feet}$ is equal to $2\text{ seconds}$ .