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 = 20,r = 0.5

Verified step by step guidance

1. Understand that the sequence {hₙ} represents the height of the ball after the nth bounce, where h₀ is the initial height before any bounce.
2. Recognize that after each bounce, the ball reaches a height that is a fixed fraction r of the previous height, which means the sequence is geometric.
3. Recall the general formula for the nth term of a geometric sequence: \(h_n = h_0 \times r^n\), where \(h_0\) is the initial term and \(r\) is the common ratio.
4. Substitute the given values into the formula: \(h_0 = 20\) and \(r = 0.5\), so the explicit formula becomes \(h_n = 20 \times (0.5)^n\).
5. This formula allows you to find 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. Understanding this helps in expressing the nth term explicitly.

Explicit Formula for Sequences

The explicit formula for a geometric sequence allows direct computation of the nth term without finding all previous terms. It is given by hₙ = h₀ * rⁿ, where h₀ is the initial term and r is the common ratio. This formula is essential to find the height after the nth bounce efficiently.

Exponents and Powers

Exponents represent repeated multiplication of a base number. In the context of geometric sequences, the common ratio is raised to the power of n to find the nth term. Understanding how to work with exponents is crucial for manipulating and interpreting the explicit formula.