5.01 Exploring sequences
Introduction
Throughout all math courses, we study and describe many different number patterns like even numbers, square numbers, triangular numbers, and many others. Here we look at how we can use expressions to describe patterns in number sequences with a combination of recursive and subscript notation.
Sequences in recursive notation
Each number in a sequence is called a term. Sequences can be finite or infinite.
Exploration
Consider the following patterns.
- How could you describe these patterns in words?
- Which patterns could be finite, and which ones could be infinite?
- If we only consider the number of objects in each step, how could we represent the patterns mathematically?
To describe a pattern, we consider what is happening from one step to the next. When we use a previous step to describe what happens in the next step, we are using a recursive pattern.
One of the most famous recursive patterns is the Fibonacci sequence.
1,\,1,\,2,\,3,\,5,\,8,\,13,\,21, \ldots
To get the next term of the sequence, we need to add the two previous terms. This is how we can describe the recursive pattern in words, but let's see how it can be represented mathematically.
The general rule for finding the nth term of the Fibonacci sequence in recursive notation is \\a_n=a_{n-2}+a_{n-1} where a_1=1 and a_2=1.
To understand the rule, we can begin by labeling each of the terms using the notation a_n where n represents the number of the term.
We are given the first two terms in the rule, so the next one we need to find is the third term, a_3. To find the third term, we are letting n=3. Plugging this into the rule, we get:
| \displaystyle a_3 | \displaystyle = | \displaystyle a_{3-2}+a_{3-1} |
| \displaystyle = | \displaystyle a_1+a_2 |
In other words, we need to add the first two terms, a_1 and a_2, to find the third term, a_3.
| \displaystyle a_3 | \displaystyle = | \displaystyle a_1+a_2 |
| \displaystyle = | \displaystyle 1+1 | |
| \displaystyle = | \displaystyle 2 |
Now, we can use this rule to find the next term, a_4. In this case, n=4.
| \displaystyle a_4 | \displaystyle = | \displaystyle a_{4-2}+a_{4-1} |
| \displaystyle = | \displaystyle a_2+a_3 | |
| \displaystyle = | \displaystyle 1+2 | |
| \displaystyle = | \displaystyle 3 |
Writing recursive rules relies on some specific notation:
Each term of a sequence can be viewed as an ordered pair, \left(n,a_n\right), where the number of the term is the input and the value of the term is the output. Since each term (output) is identified by a unique term number (input), every sequence is also a function. The domain of a sequence will be some subset of the integers.
Examples
Example 1
Consider the following sequence:
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| a_n | -7 | -3 | 1 | 5 | 9 |
Describe the recursive pattern in words.
Find the next 3 terms of the sequence.
Write the recursive rule for this sequence, t_{n+1}, including the first term, t_1.
Example 2
A sequence is defined by the rule T_{n+2}=2\left(T_{n+1}-T_{n}\right) with T_{1}=-2 and T_2=0.
Find the first 6 terms of the sequence.
Example 3
In a certain sequence, the next term is found by doubling the previous term, then subtracting one.
Write the recursive rule, a_n.
If a_1=-2, find a_5.
Example 4
Consider the graph of the pattern:
Write the recursive rule, T_{n+1}, and the initial term T_0.
Recursive rules can be written in different forms, but they represent the same sequence. For a sequence where we subtract 4 to find the next term, we can represent it in either of the following ways:
Given an initial term such as a_0=c or a_1=c,
a_n=a_{n-1}-4
a_{n+1}=a_n-4
For a recusive rule that uses two previous terms, the first two terms must be given.