1. What are two types of Sequences?
What is a sequence?
A sequence of numbers is a list, an ordering that may or may not follow some rule.
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.

2. When we are given a sequence and asked to find a specific term…
1.Determine whether the sequence is arithmetic or geometric.
2. Find the common difference (arithmetic) or the common ratio (geometric)
3. Use the appropriate formula to find the term asked for.
Arithmetic Geometric

3. We will be reviewing arithmetic
Find 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
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 sequence

4. Now use the information in the formula…don’t plug in “n” and you will have the explicit form
This is the explicit form
Distribute the -6 and simplify, change to f(n) and you will have function form
This is function form
To find the 10th term use n = 10 and simplify

5. So let’s list some steps
First find the common difference
Next list all of the unknowns
Find the explicit form by plugging in the first term and the common difference
Find the function form by simplifying the explicit form
To find the specific term replace n with the term you are looking for.

6. You try…
Find the common difference, the indicated term, write the explicit and function form of the sequence.
6,106, 206, 306,…n=15
-38,-45,-52,-59,…n=23
-16,14, 44, 74,…n=52
1406
-206
1574

7. 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.

8. Let’s look at the following sequence
Sequence and Terms
Let’s look at the following sequence n²
Do you know what the rule is for the sequence? 1, 4, 9,
16,
25,
36,
49, …,
The letter a with a subscript is used to represent function values of a sequence.
The subscripts identify the location of a term.

9. How to read the subscripts:
a term in the sequence
the prior
term
the next term

10. Example 1: Find the first four terms of the sequence:
General Term
Let’s be sure we understand what is given
+ 2 is
Each term after the first
3 times the previous term
The first term is 5
Plus 2

11. Continued… EX 1: Find the first four terms of the sequence:
Start with general term for n>1
n=1
given
n=2
n=3
n=4
Answer = 5, 17, 53, 161

12. Your turn: Ex 2: Find the next four terms of the sequence.
Start with general term for n>1
n=1
given
n=2
n=3
n=4
Answer = 3, 6, 12, 24

13. Write a recursive formula for the arithmetic sequence below
Write a recursive formula for the arithmetic sequence below. Step 1 : Make sure it is arithmetic Step 2 : Plug into the arithmetic recursive formula. Step 3 : Make sure you tell us what a1 is equal to.
Ex. 3
7, 3, -1, -5, -9, …
The common difference = -4
The first term = 7

14. Choose the recursive formula for the given sequence.
Last Example 
Choose the recursive formula for the given sequence.
Answer = C