Problem 81 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Beginning at 6: 45 A.M., a bus stops on my block every 23 minutes, so I used the formula for the \(n\) th term of an arithmetic sequence to describe the stopping time for the \(n\) th bus of the day.

Step-by-Step Solution

Answer

The statement makes sense because the bus intervals can indeed be represented as an arithmetic sequence.

Step 1: Understanding the Statement

The statement mentions that a bus stops on the block every 23 minutes starting at 6:45 A.M. The bus stopping times can be seen as an arithmetic sequence where the common difference (\(d\)) is 23 minutes. The first term (\(a_1\)) of this sequence would be the time of the first bus, which is 6:45 A.M.

Step 2: Applying the Arithmetic Sequence Formula

The formula for the \(n\)th term of an arithmetic sequence is \(a_n = a_1 + (n - 1) * d\). In this context, \(a_n\) would represent the time the \(n\)th bus of the day would arrive. We can substitute \(a_1\) with '6:45 A.M.' and \(d\) with '23 minutes' in this formula.

Step 3: Evaluation

After applying the arithmetic sequence formula, we can conclude that it makes sense to use this formula to determine the arrival time of the \(n\)th bus of the day. Each subsequent bus comes 23 minutes after the previous one, which is exactly the concept of an arithmetic sequence.

Key Concepts

Arithmetic Sequence Formula

The arithmetic sequence formula provides a straightforward way to find any term in a sequence where each term is a fixed number apart from its predecessor. The formula is given by \( a_n = a_1 + (n - 1) * d \), where

• \(a_n\) is the nth term in the sequence,
• \(a_1\) is the first term,
• \(n\) is the term number, and
• \(d\) is the common difference between the terms.

Let us clarify this with the bus schedule example from the exercise. We're given that the first bus stops at 6:45 AM. This is our first term, \(a_1\). The buses stop every 23 minutes, making this our common difference, \(d\). Should we want to find when the 10th bus of the day will arrive, we plug in these values into our formula: \(a_{10} = \) '6:45 A.M.' + \((10 - 1) * 23\) minutes.

By following these steps, it is easy to calculate the stopping time of any bus throughout the day, once we know the first bus time and the time interval between buses.

Common Difference

The common difference in an arithmetic sequence is the constant amount that each term in the sequence increases by from the previous term. It is denoted by \(d\) in our computations. In the context of our bus schedule, the common difference is \(23\) minutes, meaning each bus arrives exactly 23 minutes after the last one.

Understanding the common difference is crucial because it defines the behavior of the sequence. If every consecutive number in the sequence varies by the same amount—like the consistently spaced bus arrivals—then we are indeed dealing with an arithmetic sequence. This uniform interval makes calculating subsequent terms predictable and allows the use of the arithmetic sequence formula for efficient computation.

Sequence and Series

A sequence is a set of numbers arranged in a particular order following a certain rule. When dealing with arithmetic sequences, this rule is the addition or subtraction of the common difference to the previous term. On the other hand, a series is the sum of the terms in a sequence. While our bus schedule problem is primarily concerned with the sequence of bus arrival times, the concept of series is relevant when we are adding the terms together, possibly to find total time frames or cumulative distances.

For instance, if we wanted to know the total amount of time passed from the first bus until the tenth, we would sum the first ten terms of our bus arrival sequence. The arithmetic series can be calculated using the formula: \( S_n = \frac{n}{2} * (2a_1 + (n - 1)d) \), where

• \(S_n\) is the sum of the first n terms,
• \(a_1\) is the first term,
• \(n\) is the number of terms, and
• \(d\) is the common difference.

This formula can be applied to deduce various outcomes, like travel schedules, budget forecasts, and even population projections if the data forms an arithmetic sequence.