Arithmetic/Geometric math question

Question

You are responsible to provide tennis balls for a tournament. You know that a ball bounces like new when it is dropped and it bounces 82% of the previous height. Because of budget restriction for the tournament you need to test some used balls to make sure they bounce like new tennis balls. You drop a ball and it bounces multiple times; each bounce reaches 82% the height of the previous height.

a. Is this sequence geometric or arithmetic? Explain.
b. What are the heights of the first four bounces of a new ball if it is dropped from a height of 10 feet?
c. What is an equation that will find the nth term of this sequence?
d. Does this sequence diverge or converge? Explain.
e. What is the sum of the heights of the bounces for the first ten bounces of a new ball if it is dropped from 10 feet?

Answer 1

a. Is this sequence geometric or arithmetic? Explain.

Geometric. The change in height of bounce is a common ratio, r=0.82.

b. What are the heights of the first four bounces of a new ball if it is dropped from a height of 10 feet?

a₁ = 10

a₂ =10(.82)

    = 8.2

a₃ = 8.2(.82)
    = 6.24

a₄ =6.24(.82)

    = 5.51

a₅ =5.51(.82)

     = 4.52

c. What is an equation that will find the nth term of this sequence?

a_n = a₁(0.82)^n-1

d. Does this sequence diverge or converge? Explain.

Converges because 0.82< 1.

 

e. What is the sum of the heights of the bounces for the first ten bounces of a new ball if it is dropped from 10 feet?

S₁₀ =10[1-(0.82)¹⁰] / (1-0.82)

       = 47.92