Question

A ball is thrown upward to a height of \(h_{0}\) meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let \(h_{n}\) be the height after the nth bounce and let \(S_{n}\) be the total distance the ball has traveled at the moment of the nth bounce. a. Find the first four terms of the sequence \(\left\\{S_{n}\right\\}\) b. Make a table of 20 terms of the sequence \(\left\\{S_{n}\right\\}\) and determine \(a\) plausible value for the limit of \(\left\\{S_{n}\right\\}\) $$h_{0}=20, r=0.75$$

Short answer

Short Answer: The first four terms of the sequence {Sₙ} are approximately 35, 47.50, 56.25, and 62.19, representing the total distance traveled by the ball after 1, 2, 3, and 4 bounces, respectively. Analyzing a table of 20 terms of the sequence and the pattern of the terms as they increase, we can determine that the plausible value for the limit of the sequence {Sₙ} is approximately 80.

Step-by-step solution

Calculating height after each bounce

We are given that \(h_{0} = 20\) and \(r = 0.75\), we can calculate the height after each bounce using the formula: $$h_{n} = h_{0} * r^{n}$$

Finding the total distance traveled after each bounce

To calculate the total distance traveled after each bounce, we can use the following formula: $$S_{n} = h_{0} + 2\sum_{i=1}^{n} h_{i}$$ Now, let's find the first four terms of the sequence \(\left\\{S_{n}\right\\}\):

Calculating \(\left\\{S_{1}\right\\}\):

The total distance traveled after the first bounce can be calculated as follows: $$S_{1} = h_{0} + 2(h_{1})$$ $$S_{1} = 20 + 2(20 * 0.75^{1}) \approx 35$$

Calculating \(\left\\{S_{2}\right\\}\), \(\left\\{S_{3}\right\\}\), and \(\left\\{S_{4}\right\\}\)

Using the same formula, we can find the distances traveled by the ball after the 2nd, 3rd, and 4th bounces: $$S_{2} = 20 + 2(20 * 0.75^{1}+20 * 0.75^{2}) \approx 47.50$$ $$S_{3} = 20 + 2(20 * 0.75^{1}+20 * 0.75^{2}+20 * 0.75^{3}) \approx 56.25$$ $$S_{4} = 20 + 2(20 * 0.75^{1}+20 * 0.75^{2}+20 * 0.75^{3}+20 * 0.75^{4}) \approx 62.19$$ b) Making a table of 20 terms of the sequence \(\left\\{S_{n}\right\\}\) and determine a plausible value for the limit of \(\left\\{S_{n}\right\\}\):

Generating the Table of 20 terms

Using the formula for \(S_{n}\), we can generate a table of 20 terms for the sequence \(\left\\{S_{n}\right\\}\). Observe the pattern of the terms and their values as they increase.

Determining the Limit of \(\left\\{S_{n}\right\\}\)

Using the analysis of the table, we can determine that the limit of the sequence \(\left\\{S_{n}\right\\}\) is: $$S_{n}\rightarrow \frac{h_{0}(1+r)}{1-r} = \frac{20(1+0.75)}{1-0.75}\approx 80$$

Key concepts

Understanding Geometric Sequences

A geometric sequence is a series of numbers where each term is found by multiplying the previous term by a constant called the common ratio, denoted as \( r \). In the context of our exercise, the height to which a ball rebounds after each bounce forms a geometric sequence. Starting with a height \( h_0 = 20 \) meters, each subsequent height \( h_n \) can be calculated using the formula: \[ h_n = h_0 \times r^n \] Here, \( r = 0.75 \) represents the fraction of height retained after each bounce. This means, in practical terms, the ball reaches 75% of its previous height each time it bounces. The first four terms for heights are derived by substituting \( n \) with 1, 2, 3, and 4. This method easily allows us to predict subsequent heights, which is fundamental when dealing with patterns or behaviors described by geometric sequences.

Exploring the Convergence of Series

When discussing series in mathematics, convergence refers to the behavior of a sequence as the number of terms increases indefinitely. Specifically, a convergent series reaches a limit. Let's revisit the sequence \( \{S_n\} \), which represents the total distance the ball has traveled after each bounce. Each \( S_n \) is given by \[ S_n = h_0 + 2 \sum_{i=1}^{n} h_i \] As \( n \) increases, the height of each bounce diminishes due to the factor \( r < 1 \). The distance accumulates at a decreasing rate leading us to analyze the series' convergence. The exercise solution provides a way to calculate the limit as \[ S_n \rightarrow \frac{h_0(1+r)}{1-r} \] Plugging in the given values, the limit is approximately 80 meters. This theoretical limit suggests that as the bounces continue indefinitely, the total distance approaches this value, indicating convergence.

Delving into Recursive Formulas

Recursive formulas are a powerful tool in mathematics for defining a sequence where each term is derived from the previous ones. In the ball bounce problem, recursive reasoning helps describe the change in height and cumulative distance with each bounce. While the problem primarily utilizes explicit formulas for direct calculation, recursive thinking allows deeper understanding and simpler computation in some scenarios. The formula for height \( h_n = h_{n-1} \times r \) recursively describes each height based on its predecessor. Though less emphasized here, recursion smoothly extends to more complex sequences, bridging the gap between initial data and their long-term implications. Understanding recursive structures empowers students to grasp processes that evolve over time through iterative steps.