Chapter 7.4, Problem 5GP

Solution

To calculate : the distance a pendulum swings if the pendulum travels 10 inches on its first swing. On each succession the pendulum travels $80\%$ of the distance of the previous swing.

The total distance a pendulum swings is $5$ inches.

Given information :

The pendulum travels 10 inches on its first swing. On each succession the pendulum travels $80\%$ of the distance of the previous swing.

Formula used : The sum of infinite geometric series with first term $a_1$ and common ration $r$ is calculated using the formula $S=\frac{a_1}{1-r}$ .

Here, $\left|r\right|<1$ . If $\left|r\right|\ge 1$ , then the series has no sum.

Calculation :

It can be observed that the person swings by one swing and every next swing is equal to $80\%$ of the distance of the previous swing.

On the first swing the person travels distance of $10$ inches.

So,

  $a_1=10$

On the second swing, the person travels $80\%$ of the distance of the first swing.

So,

  $a_2=0.8a_1$

Similarly,

  $a_3=0.8a_2$

It can be seen that, the sequence forms a geometric sequence with value of $r$ calculated as follows.

  $\begin{array}{c}r=\frac{a_2}{a_1}\\ =\frac{0.8a_1}{a_1}\\ =0.8\end{array}$

Observe all the terms as follows.

  $\begin{array}{c}{a}_2=0.8a_1\\ =0.8\left(10\right)\\ =10\left(0.8\right)\end{array}$

  $\begin{array}{c}{a}_3=0.8a_2\\ =0.8\cdot 10\left(0.8\right)\\ =10{\left(0.8\right)}^2\end{array}$

  $\begin{array}{c}{a}_4=0.8a_3\\ =0.8\cdot 10{\left(0.8\right)}^2\\ =10{\left(0.8\right)}^3\end{array}$

The general rule for the given geometric sequence is formed as $a_n=10{\left(0.8\right)}^{n-1}$ .

For the given series the value of $a_1$ is $10$ and $r$ is $0.8$ .

The sum of series can only exist if the common ratio is $\left|r\right|<1$ .

  $\begin{array}{c}\left|0.8\right|<1\\ 0.8<1\end{array}$

Then the sum of infinite geometric series is calculated using the formula $S=\frac{a_1}{1-r}$ .

  $\begin{array}{c}S=\frac{10}{1-0.8}\\ =\frac{10}{0.2}\\ =5\end{array}$

Thus, the total distance a pendulum swings is $5$ inches.