In this lesson, our instructor Vincent Selhorst-Jones teaches Intro to Sequences. Youll learn the definition of a sequence, including a infinite sequence and finite sequence. Then, Vincent will explain the formula for the nth term, define a sequence recursively, and then go over sequences and patterns. Youll also learn some good tips for finding patterns. The lesson ends with six fully worked-out examples for extra practice.
A sequence is an ordered list of numbers. We can write out a sequence as
a1, a2, a3, a4, …, an, …
We call each of the entries in the sequence a term. Above, a1 is the first term, a2 is the second term, and so on. Any symbol can be used to denote the sequence, while the subscript (small number to the right) tells us which
term it is in the sequence.
If a sequence goes on forever without stopping, it is called an infinite sequence. Most of the sequences we will work with will be infinite sequences. If a sequence does stop, it is called a finite sequence.
a1, a2, a3, a4, …, ak
We call the number of terms in a finite sequence its length. The length of the above sequence is k.
If we know a formula for the nth term (this is also called the general term), we can easily find any term. Plug the appropriate value for n into the formula, then work out what that term is. For example, if we want to find the
seventh term, we would plug in n=7.
A sequence can also be defined recursively: each term is based on what came before. The sequence is built on a recursion formula that shows how each term is built from preceding terms. To use a recursion formula, we need a "starting" place
before we can make a sequence. This is called the initial term (or terms, if multiple are needed).
Given a recursion formula and initial term(s), it can be possible to find a formula for the nth term. Similarly, it can be possible to transform an nth term formula into a recursion formula and initial term(s). Still, there is no
guarantee we can do this. Sometimes it will be easy, sometimes hard, and sometimes impossible.
Very, very often you will be given the first few terms of a sequence and told to either give more terms, or figure out a formula for the nth term. To do this, you will have to find some pattern in the sequence, then exploit it.
When trying to recognize a pattern in a sequence, try to think in terms of how to get from one term to the next. Establish a hypothesis by looking at a1→ a2, then test it against a2→ a3, a3→
a4, and any other terms given. Once you've figured out the pattern, it's easy to find further terms in the sequence. Finding a formula for the nth term can be tricky, though. Think carefully about how you can put the pattern in an
equation, then make sure to check some terms after you create the formula.
When trying to find the pattern in a sequence, there are a variety of common pattern types that appear. Here are some important ones to keep in mind:
Addition/Subtraction: add k every term.
Mutliplication/Division: multiply by k every term.
Squares (n2): 1, 4, 9, 16, 25, 36, 49, …
Cubes (n3): 1, 8, 27, 64, 125, 216, …
Factorials (n!): 1, 2, 6, 24, 120, 720, …
Alternating Signs: Alternating signs can created by (−1)n+1 or (−1)n.
If most of the terms in a sequence are presented in a certain format, like fractions, try to figure out a way to put all the terms in that format. It can be easier to see patterns if everything is in the same format. Furthermore, if the format can be
clearly broken into multiple parts (in a fraction [¯]/[¯], we can break it into numerator and denominator), it can help to figure
out patterns for each part separately.
It can sometimes help to write the number of the term above or below each term (n=1, n=2, etc). This helps you keep track of numerical location, which often makes it easier to identify patterns.
In the end, there is no one way to identify all patterns. Try to take a broad view of the sequence and look for repetitions or similarities to other patterns you've seen. If you still can't figure it out, see if there's an alternative way to write the
terms out. Persevere, and be creative.
Introduction to Sequences
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.