SEQUENCESA sequence is a function whose domain in the set of positive integers.

So if I gave you a function but limited the domain to the set of positive integers and you substituted them in beginning with 1 then 2 then 3 etc. you would generate a sequence. 23 xxf

,13,10,7,4,1This is a sequence generated by putting 1, 2, 3, 4, 5 … in the function above.

,13,10,7,4,1

We can look at the sequence and see a pattern:

+3 +3 +3 +3

For a sequence we usually use a little different notation than for functions. We use the letter a for a term in the sequence and subscript it with the term number. The term number is also the number that you substitute in the formula for a sequence to get that term’s value.

23 nanThis would be how we’d express the sequence above.

If I wanted to know the 10th term I’d put 10 in the sequence formula (often called the term generator)

28210310 a

12 n Write the first five terms of the sequence.

21121 a

2,

51222 a

5,

101323 a 17142

4 a

261525 a

10, 17, 26,

Can you see the pattern and guess the next term?

37,

,16,8,4,2,1 Can you figure out the formula for this sequence?

We can see that each term is twice as big as the next except that the terms alternate signs. Alternating signs in a sequence mean there must be a negative sign to a power because odd powers would be negative and even powers would be positive.

Let’s make a guess on what it might look like and then modify our guess until we have it right.

n2

Let’s sub in 1 then 2 then 3 and see what we get.

,16,8,4,2

Looks good but we lost the first term. Let’s modify our guess.

12 n

,16,8,4,2,1

Another way to define a sequence is using what we call a recursive formula. Basically it is the rule of what you do to a term to get the term after it.

,13,10,7,4,1

+3 +3 +3 +3

We saw the pattern in the first sequence:

So to get to the next term we add three. The recursive formula would look like this:

31 nn aa

to get the nth term take the n-1 term (the term right before it) and add 3

Let’s write the recursive formula for the other sequence we had:

,16,8,4,2,1 To get to the next term you multiply by negative two.

12 nn aa

With this formula, to get to the next term you must substitute the term before it. While this formula basically shows you the pattern, you can’t easily get at the 10th term without knowing the 9th term.

910 2aa You’d need the 9th term here to find the 10th term

= 1

Let’s try generating a sequence from a recursive formula:

tells us the first term

take the term number

Use this formula to find the second term.

1122 22 aaa

would be when n = 2

1Find the next 3 terms.

1, 1, 2, 2, 3, …

11 ;1 nn anaa

subtract the term before

2133 23 aa2244 34 aa3255 45 aa

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au