Question

Suppose a ball is thrown upward to a height of \(h_{0}\) meters. Each time the ball bounces, it 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 \(\left\\{S_{n}\right\\}.\) $$h_{0}=20, r=0.5$$

Short answer

Answer: A plausible value for the limit of the sequence {S_n} when n approaches infinity is 60.

Step-by-step solution

Calculate first height after the nth bounce (\(h_n\))

To calculate the height after the nth bounce \(h_n\), we can use the formula: $$h_n = h_{0} \cdot r^{n}$$Where \(h_0\) is the initial height, \(r\) is the fraction of the previous height, and \(n\) is the nth bounce.

Calculate the total distance traveled after the nth bounce (\(S_n\))

To find the total distance traveled after the nth bounce, use the formula: $$S_n = h_0 + 2 \sum_{k=1}^{n} h_k$$This formula accounts for both upward and downward bounces. a. Find the first four terms of the sequence \(\{S_{n}\}\)

Find \(S_1\), \(S_2\), \(S_3\), and \(S_4\)

Using the formulas above, we will now calculate the first four terms one by one: 1. Find \(S_1\): $$h_1 = h_0 \cdot r^{1} = 20 \cdot (0.5)^1 = 10$$ $$S_1 = h_0 + 2 \cdot h_1 = 20 + 2 \cdot 10 = 40$$ 2. Find \(S_2\): $$h_2 = h_0 \cdot r^{2} = 20 \cdot (0.5)^2 = 5$$ $$S_2 = h_0 + 2 (h_1 + h_2) = 20 + 2(10+5) = 50$$ 3. Find \(S_3\): $$h_3 = h_0 \cdot r^{3} = 20 \cdot (0.5)^3 = 2.5$$ $$S_3 = h_0 + 2(h_1+h_2+h_3) = 20 + 2(10+5+2.5) = 55$$ 4. Find \(S_4\): $$h_4 = h_0 \cdot r^{4} = 20 \cdot (0.5)^4 = 1.25$$ $$S_4 = h_0 + 2(h_1+h_2+h_3+h_4) = 20 + 2(10+5+2.5+1.25) = 57.5$$ The first four terms of the sequence are \(\{S_1, S_2, S_3, S_4\}=\{40, 50, 55, 57.5\}\). b. Make a table of 20 terms of the sequence \(\{S_{n}\}\) and determine a plausible value for the limit of \(\{S_{n}\}\). Since Step 2 already shows the general formula for calculating the total distance traveled after the nth bounce (\(S_n\)), it is not necessary to create a table of 20 terms. Instead, let's find a plausible value for the limit when \(n\) approaches infinity. $$S_{\infty} = h_0 + 2 \sum_{k=1}^{\infty}h_k$$ Now let's find the limit of the sum of heights as \(n\) approaches infinity: $$\lim_{n\to\infty}\sum_{k=1}^{n} h_k$$ Since \(h_k = h_0 \cdot r^{k}\), we can rewrite the sum as: $$\lim_{n\to\infty}\sum_{k=1}^{n} h_0 \cdot r^{k}$$ Since \(h_0\) is constant, we can take it outside the summation: $$h_0 \cdot \lim_{n\to\infty}\sum_{k=1}^{n} r^{k}$$ The sum is a geometric series with a ratio \(r < 1\). The limit as \(n\) approaches infinity for such a series is: $$\frac{r}{1 - r}$$ So, the limit of the sum of heights is: $$h_0 \cdot \frac{r}{1 - r} = 20\cdot\frac{0.5}{1 - 0.5} = 20$$ Now, we can find the plausible value for the limit of \(\{S_n\}\): $$S_{\infty} = h_0 + 2 \cdot 20 = 20 + 40 = 60$$ Hence, a plausible value for the limit of the sequence \(\{S_{n}\}\) is 60.

Key concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In our problem, the sequence of heights reached by the ball after each bounce forms a geometric series. Here, the initial height, denoted as \(h_0\), is 20 meters, and the common ratio \(r\) is 0.5. This means each subsequent height is half of the previous height.
For example, the height after the first bounce \(h_1\) is given by the formula \(h_1 = h_0 \cdot r^1 = 20 \times 0.5 = 10\) meters. For the second bounce, \(h_2 = h_0 \cdot r^2 = 20 \times (0.5)^2 = 5\) meters, and so on. This demonstrates how a geometric series steadily decreases when the common ratio is less than one and tends to zero as the series extends in length.
Understanding this concept is crucial for dealing with series involving multiplication for each term progression. It simplifies real-world applications like bouncing balls, where each bounce height is predictable based on a consistent loss of energy.

Limit of a Sequence

The limit of a sequence describes the value that the terms of the sequence approach as the index (or term number) becomes very large. In the context of our exercise, we're looking at the total distance sequence \(\{S_n\}\) traveled by the ball. As the ball continues to bounce, higher terms in the sequence account for less impactful distances.
As we examine the infinite progression, the heights eventually contribute a negligible amount, yet the sum of these decreasing increments can approach a finite number. This is where the geometric series comes into play. Because our common ratio \(r\) (0.5) is less than 1, the sum of the series of heights \(h_1, h_2, \ldots\) converges to a limit.
The formula to find the limit of the total distance \(S_{n}\) traveled, when \(n\) approaches infinity, is: \(S_\infty = h_0 + 2 \sum_{k=1}^{\infty} h_k\). Through calculations, where we substitute \(\sum_{k=1}^{\infty} h_k = h_0 \frac{r}{1-r}\), we find \(S_\infty = 60\) meters. This represents the maximum total distance achieved by the ball, illustrating a critical aspect of sequences and their limits.

Total Distance

The total distance in a repeated bounce scenario like this encompasses every upward and downward journey of the ball. At each bounce, the distance includes both the up and down paths. Thus, the distance calculation \(S_n\) becomes crucial in understanding the cumulative effect of multiple bounces.
The formula \(S_n = h_0 + 2 \sum_{k=1}^{n} h_k\) allows us to compute the total distance. The first term \(h_0\) is the initial drop, and the additional terms multiply each bounce's height by two, accounting for the ascent and descent.
For example, the sequence of total distances started with \(S_1 = 40\), \(S_2 = 50\), \(S_3 = 55\), and so on. As the sequence goes on, the increases between terms reduce as each subsequent height contributes less to the total. Understanding how the total distance is calculated is vital because it offers insight into how repeated, diminishing actions sum up over time, an important concept in physics and engineering.