57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.


b. Find an explicit formula for the nth term of the sequence {hₙ}.


h₀ = 30,r = 0.25

Verified step by step guidance

1. Understand the problem: The height after each bounce forms a geometric sequence where the first term is the initial height \(h_0\) and the common ratio is the rebound fraction \(r\).
2. Recall the formula for the nth term of a geometric sequence: \(h_n = h_0 \times r^n\), where \(h_n\) is the height after the nth bounce.
3. Identify the given values: \(h_0 = 30\) meters and \(r = 0.25\).
4. Substitute the given values into the formula to write the explicit formula for the nth term: \(h_n = 30 \times (0.25)^n\).
5. This formula allows you to calculate the height after any number of bounces by plugging in the value of \(n\).

Key Concepts

Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant ratio. In this problem, the heights after each bounce form a geometric sequence with initial term h₀ and common ratio r, representing the fraction of height retained after each bounce.

Explicit Formula for Geometric Sequences

The explicit formula for the nth term of a geometric sequence is given by hₙ = h₀ * rⁿ, where h₀ is the initial term and r is the common ratio. This formula allows direct calculation of any term in the sequence without needing to find all previous terms.

Application of Sequences to Physical Contexts

Modeling real-world phenomena like bouncing balls with sequences helps translate physical behavior into mathematical terms. Here, the height after each bounce decreases by a fixed fraction, illustrating how sequences can describe repeated processes with diminishing values.