Or

Author:
c o

To introduce mathematical sequences to the learner and to become familiar with the concepts of sequences, recursive formula, and factorial notation.

Sequences are introduced by looking at an example. The general term is then covered before moving on to a section on recursive formula and the Fibonacci sequence. Finally, factorial notation is defined and examples of sequences using factorials are given.

Tutorial

A simple and intuitive idea provides the foundation for our understanding of sequences: **a sequence is just a list of numbers**. The numbers on the list are called the **terms** of the sequence, and most often we are interested in sequences only when their terms elicit some pattern. When such a pattern exists, the sequence is said to be defined by a **rule**. To make these ideas clearer, let us consider a real-live sequence:

We can see that this is just a list of numbers. We also see that a pattern exists between the terms. Expressing the pattern as a rule, and doing so in plain English, we could say that to get from one member of the sequence to the next, we just add * 3*.

Sequences can be either **finite** or **infinite** in length. The sequence above is infinite - we can keep adding 3 to find successive terms to our hearts' content. A finite sequence, on the other hand, is made up of finitely many terms.

In order to write sequences down more succinctly, we employ the concept of a **general term**. The general term of a sequence is usually written as a variable with a subscript, for example * x_{n}*, where

So, for the first term, * x_{1}*, we have

We often visualize a sequence using a table, like this

The table can sometimes help us see the pattern between * n* and

Looking back at our example sequence, consider again the rule that defines its terms. To get from the first term to the second, we add * 3* to the first. To get from the second term to the third, we add

What does this mean? It means that if we want to find * x_{n}*, we first look at the previous term

Recursive formulas have the advantage of providing a recipe for any term in the sequence, just so long as we already know the terms that came before it. The downside is that finding all the terms leading up to the term that we're interested can take a long time.

Using the above recursive formula we can generate the whole sequence:

**x _{1} = 4 **

**x _{2} = x_{1} + 3 = 4 + 3 = 7 **

**x _{3} = x_{2} + 3 = 7 + 3 = 10**

**x _{4} = x_{3} + 3 = 10 + 3 = 13**

**...**

It seems like using a recursive formula is tedious. Why not just use the regular formula * x_{n} = 3n + 1* instead? For the example we've been working with so far, the regular formula probably is the best way to find a particular term in the sequence. There are some sequences, however, for which a recursive formula may be a simpler. An example is the famous Fibonacci sequence. Here is what it looks like:

How might we write a general term for this sequence? Writing the * n^{th}* term as a function of

If we call the general term * f_{n}*, then we see that it is a sum of the two preceding terms. In the graphic above, the third term,

Using this formula, can we find the eighth Fibonacci number, * f_{8}*? Of course! Letting

A recursive formula is just a way to express the idea that one term in a sequence is determined by the some of the terms that precede it. Usually when we write a recursive formula, we will also define the **initial terms **of the sequence. We have already seen two examples using initial terms; one, when we set * x_{1} = 1* in our first example; and another, when we set the first two Fibonacci numbers

We are about to see a few more examples of sequences, but first we introduce the concept of factorials. Let n be any natural number, then the factorial of n, written n!, is the product of n and all the natural numbers that come before it. For example, when n is 5, then we have

In general,

Also, by convention we define * 0! = 1*.

Using what we have just learned about recursive formulas, we can see that * n!* admits a recursive definition. Letting our initial term be

*Example 1*

Let the general term of our sequence be defined by * x_{n} = (n+2)/n!* What is the fourth term of this sequence? We can find it by setting

*Example 2*

Now let the general term be defined * x_{n} = n! - 1*. What are the first five terms of this sequence?

**x _{1} = 1! - 1 = 1 - 1 = 0**

**x _{2} = 2! - 1 = (2 • 1) - 1 = 1**

**x _{3} = 3! - 1 = (3 • 2 • 1) -1 = 6 - 1 = 5 **

**x _{4} = 4! - 1 = (4 • 3 • 2 • 1) - 1 = 24 - 1 = 23**

**x _{5} = 5! - 1 = (5 • 4 • 3 • 2 • 1) - 1 = 120 - 1 = 119**