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((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(1-r^n)/(1-r) \\ =5\cdot (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(1-r^n)/(1-r) \\ =5\cdot(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.