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

Answer:

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

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

Step-by-step explanation:

If each curved path has 52% of the previous height this means that $\frac{a_{n+1}}{a_n} = 0.52$

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 $a_6$

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