Sequences

A sequence of numbers is a list, an ordering that may or may not follow some rule.

What is a sequence?

What are two types of Sequences?

• Arithmetic Sequence:

• Geometric Sequence:

Is a sequence in which each term is equal to the previous term plus a constant. This constant is called the common difference.

Is a sequence in which each term is equal to the previous term multiplied by a constant. This constant is called the common ratio.

When we are given a sequence and asked to find a specific term…

1.Determine whether the sequence is arithmetic or geometric.

Arithmetic Geometric

1 1nA a n d 11

nnA a r

3. Use the appropriate formula to find the term asked for.

2. Find the common difference (arithmetic) or the common ratio (geometric)

We will be reviewing arithmeticFind the common difference, find the indicated term, write the equation in both function and

explicit form

Let’s look at a sequence.

12, 6, 0, -6, . . . Find 10a

Write down every thing you need in the formula.

n =10 because we are looking for the 10th term

d = -6 because we are adding -6 to each term

= 12 because it is the first term of the sequence1a

Now use the information in the formula…don’t plug in “n” and you will have the explicit form

1 1nA a n d

12 ( 1)( 6)nA n This is the explicit form

To find the 10th term use n = 10 and simplify

( ) 12 6 6f n n

Distribute the -6 and simplify, change to f(n) and you will have function form

( ) 18 6f n n This is function form

(10) 18 6(10)f (10) 42f

So let’s list some steps

1. First find the common difference

2. Next list all of the unknowns

3. Find the explicit form by plugging in the first term and the common difference

4. Find the function form by simplifying the explicit form

5. To find the specific term replace n with the term you are looking for.

You try…

Find the common difference, the indicated term, write the explicit and function form of the

sequence.

1. 6,106, 206, 306,…n=15

2. -38,-45,-52,-59,…n=23

3. -16,14, 44, 74,…n=52

1406

-206

1574

What is a recursive sequence?

Definition: A recursive sequence is the process in which each step of a pattern is dependent on the step or steps before it.

Recursion Formulas:A recursion formula defines the nth term of a sequence as a function of the previous term. If the first term of a sequence

is known, then the recursion formula can be used to determine the remaining terms.

Sequence and TermsLet’s look at the following sequence

1, 4, 9, 16, 25, 36, 49, …,

1a 2a 3a 4a 6a5a7a

The letter a with a subscript is used to represent function values of a sequence.

The subscripts identify the location of a term.

Do you know what the rule is for the sequence?n²

How to read the subscripts:

na1na 1na

a term in the

sequence

the priorterm

the next term

Example 1: Find the first four terms of the sequence:

1 5a 13 na

The first term is 5

Each term

after the first

na

13 2n na a 1 5a

+ 2is

3 times the

previous term

Plus 2

Let’s be sure we understand what is given

General Term

Continued…EX 1: Find the first four terms of the sequence:

1 5a

2 2 13 2a a

1 5a 13 2n na a

13 2a 3(5) 2 15 2 17

3 3 13 2a a 23 2a 3(17) 2 51 2 53

4 4 13 2a a 33 2a 3(53) 2 159 2 161

n=1

n=3

n=2

n=4

given

Start with general term for n>1

Answer = 5, 17, 53, 161

Your turn: Ex 2: Find the next four terms of the sequence.

1 3a

2 2 12a a

1 3a 12n na a

12a 2(3) 6

3 3 12a a 22a 2(6) 12

4 4 12a a 32a 2(12) 24

given

Start with general term for n>1

Answer = 3, 6, 12, 24

n=1

n=3

n=2

n=4

Write a recursive formula for the arithmetic sequence below.

Step 1 : Make sure it is arithmeticStep 2 : Plug into the arithmetic recursive formula.Step 3 : Make sure you tell us what a1 is equal to.

Ex. 37, 3, -1, -5, -9, …

The common difference = -4

1

Arithmetic

n na a d

The first term = 71 4n na a

1 7a

Last Example

Choose the recursive formula for the given sequence.

Answer = C