Question

A pendulum swings in a arc at a slower rate of speed. The distance can be written as a sequence 5,2.5, 1.25,… Answer each of the following questions.

Answer 1

Step 1

Use a calculator to find the first term that equals 0 to the nearest hundredth-thousand

The first term that equals zero to the nearest hundred thousandth is; n= 21

$The\text{ 21st term \lparen a}_{21})=0.000004768$

Step 2

The pendulum will not stop mathematically because the nth term is given as;

$a_n=5(\frac{1}{2})^{n-1}$

Can approach zero but will never be zero

Step 3

At 10 swings the total distance traveled will be;

$\begin{gathered} S_n=a_1\frac{1-r^n}{1-r} \\ =5\cdot \frac{1-0.5^{10}}{1-0.5} \\ =9.990234375 \end{gathered}$

At 50 swings the total distance traveled will be;

$\begin{gathered} S_n=a_1\frac{1-r^n}{1-r} \\ =5\cdot\frac{1-0.5^{50}}{1-0.5} \\ =10 \end{gathered}$

Step 4

My observation and conclusion are that the Pendulum swings at distances that follow a geometric progression and the total distance traveled will be about 10.