In this section, we will explore the following questions.
What is a sequence?
How can we use discrete sequences to describe real-world phenomena?
What is the difference between a recursive definition and an explicit formula for a discrete sequence?
We begin our study with an exploration of sequences.
Definition4.1.1.
A sequence is an ordered list of numbers, called terms.
That’s it! Typically, we explore sequences which continue forever; these are generally called infinite sequences. Such sequences are typically described using a mathematical rule of some sort. Let’s look at an example.
Example4.1.2.
Joris is a collector of board games. His collection currently has 37 games, and each year he budgets enough to acquire 8 new games. We’ll use the notation \(G_t\) to describe how many games Joris has in year \(t\text{,}\) where we consider \(G_0\) (that is, in year \(t=0\)) to be the number of games he currently has. Thus, \(G_0 = 37\text{.}\) We would also expect \(G_1 = 37 + 8 = 45\text{,}\) and \(G_2 = 45 + 8 = 53\text{.}\) We say that \(37, 45, 53\ldots\) are the first three terms of the sequence.
This way of describing the number of games Joris has is known as a recursive description, where each term depends on previous terms. That is, the recursive description is given by the formula
Since the sequence above only allows for whole number inputs, it cannot estimate the number of board games Joris will have in 1.3 years, or \(\pi/4\) years; the system’s prediction changes by 8 as the elapsed time changes from 1 to 2, 5 to 6, and so on. This makes the sequence discrete.
We also note the recursive nature of the definition of the sequence in Example 4.1.2. But there are other ways to describe sequences, and we turn to those now.
Activity4.1.3.
A recursive definition is nice because it is reasonably intuitive and fits with our usual understanding of how things change over time: they start from where they are now, and then change a little every so often. But perhaps we’d like to know how long it will take Joris to accumulate 500 games.
Clearly describe in 2-3 sentences a process for using the recursive definition in Example 4.1.2 that would allow you to determine how many years it would take Joris to accumulate 500 board games. Please do not use this process to answer the question!
In 2-3 sentences, describe the disadvantage in the process you used in Question 1 for finding how long it would take Joris to accumulate 500 games.
What we want is a explicit formula for \(G_t\text{.}\) That is, we want a formula that allows us to plug in a value for \(t\) that will give us the number of games \(G_t\) that Joris has in year \(t\) without having to know any of the previous terms in the sequence. Find such a formula, and compute the first three terms to convince your group that your formula produces the same sequence as (4.1.1).
The sequence described in Activity 4.1.3 is known as a linear sequence. This is because, like a line, it grows at a constant rate. Linear growth is extremely useful because it is straightforward to understand and apply. However, it has its shortcomings as well.
Activity4.1.4.
Lila is 6 years old, and is 43.5 inches tall. Her parents are told to expect her to grow at approximately 2.25 inches/year. Let \(H_t, t\ge 0\) denote Lila’s height \(t\) years from now.
Predict Lila’s height when she turns 8.
Give a recursive description for Lila’s height.
Give an explicit formula for Lila’s height.
How tall will Lila be when she is 50? Give your answer in feet (remember: there are 12 inches in a foot).
In 2-3 sentences, clearly articulate at least one shortcoming of extrapolating using linear models.
Conclusion.
In this section, we explored the idea of a (discrete) sequence. A sequence is just a list of numbers, and we considered sequences which are (in theory, at least) infinite. We saw two ways of describing a sequence other than just listing the numbers in order: recursively, and with an explicit formula. The recursive definition fits with our intuition about how quantities change over time, but it can be tedious to calculate terms that appear late in the sequence, as you need to calculate every term leading up to the one in which you’re interested. Explicit formulas, on the other hand, give us shortcuts to calculating any term we want, but can often be hard to find and can obscure the actual behavior of the sequence.
Next time, we’ll take a look at the ways in which we can combine multiple sequences to create systems. These systems can be used to describe the behavior of real-world interactions. In particular, we’ll explore a basic predator-prey model (involving two populations, the predators and the prey), and then a system of three sequences that describes the basic dynamics of the spread of a disease.
ExercisesExercises
1.
Find an explicit formula for each of the following.
\(\displaystyle 2, 5, 8, 11, 14, \ldots\)
\(\displaystyle 50, 43, 36, 29, \ldots\)
2.
The first three terms in a sequence are listed below as numbers of dots. Determine a recursive description for the sequence. If possible, determine an explicit formula for the sequence.
Figure4.1.5.
3.
The first four terms in a sequence are listed below as numbers of dots. Determine a recursive description for the sequence. If possible, determine an explicit formula for the sequence.
