Question

You drop a ball from a height of 0.5 meters. Each curved path has 52% of the height of the previous path. a. Write a rule for the sequence using centimeters. The initial height is given by the term n = 1. b. What height will the ball be at the top of the sixth path?

Answer 1

a)
`a_n = 50 (0.52) ^ {n-1}`

b)
`a_6 = 50 (0.52) ^ 5 = 1.90\ cm`

Explanation:

If each curved path has 52% of the previous height this means that
`(a_(n+1))/(a_n) = 0.52`

Then the radius of convergence is 0.52 and this is a geometric series.

The geometric series have the form:

`a_n = a_1 (r) ^ {n-1}`

Where

`a_1` is the first term of the series and r is the radius of convergence.

In this problem

`a_1 = 0.5` meters = 50 cm

`r = 0.52`

a) Then the rule for the sequence is:

`a_n = 50 (0.52) ^ {n-1}`

b) we must calculate
`a_6`

`a_6 = 50 (0.52) ^ 6-1\\\\a_6 = 50 (0.52) ^ 5 = 1.90\ cm`