Question

A pendulum swings 80 cm on its first​ swing, 76 cm on its second​ swing, 72.2 cm on its third​ swing, and 68.59 cm on its fourth swing. Complete parts a and b below.
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Part 1
a. If the pattern​ continues, what explicit formula can be used to find the distance of the n Superscript th​ swing?
a Subscript nequals
  
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Answer 1

To find the explicit formula for the distance of the \( n \)-th swing of the pendulum, we first need to look at the distances swung over each swing.

1. First swing: \( 80 \) cm
2. Second swing: \( 76 \) cm
3. Third swing: \( 72.2 \) cm
4. Fourth swing: \( 68.59 \) cm

From these distances, we can observe a pattern. The distances appear to decrease with each swing. We can calculate the differences between consecutive swings:

- From 1st to 2nd: \( 80 - 76 = 4 \) cm
- From 2nd to 3rd: \( 76 - 72.2 = 3.8 \) cm
- From 3rd to 4th: \( 72.2 - 68.59 = 3.61 \) cm

It looks like the distances are decreasing by a pattern where the difference itself is not consistent, but appears to decrease by a percentage.

To identify a general formula, we can calculate the ratio between consecutive swings:

- Ratio from 1st to 2nd: \( \frac{76}{80} = 0.95 \)
- Ratio from 2nd to 3rd: \( \frac{72.2}{76} \approx 0.95 \)
- Ratio from 3rd to 4th: \( \frac{68.59}{72.2} \approx 0.95 \)

If the distance decreases by approximately \( 5\% \) each time, this hints at an exponential decay model.

This can be modeled as:
\[
a_n = a_1 \cdot r^{(n-1)}
\]

Where:
- \( a_n \) is the distance of the \( n \)-th swing
- \( a_1 = 80 \) cm (the first swing)
- \( r \approx 0.95 \) (the reduction factor)

Therefore, the explicit formula for the distance of the \( n \)-th swing can be written as:
\[
a_n = 80 \cdot (0.95)^{(n-1)}
\]

So the answer you should enter for part (a) is:
\[
a_n = 80 \cdot (0.95)^{(n-1)}
\]