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4-7 Arithmetic Sequences

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4-7 Arithmetic Sequences

Objective: To identify and extend patterns in sequences and represent in function notation

- Getting Ready on page 276. Solve IT

- A sequence is an ordered list of numbers that often forms a pattern.
- Each number in the list is called a term of a sequence.
- Some sequences can be modeled with a function rule so that you can extend the sequence to any value.

- Example 1: What are next two terms?
- A. 5, 8, 11, 14 . . .
- B. 2.5, 5, 10, 20 . . .

- An arithmetic sequence: the difference between consecutive is a constant term, which is called the constant difference.
- Example 2: Is it an arithmetic sequence?
- A. 3, 8, 13, 18 . . .
- B. 6, 9, 13, 17 . . .

- Sequences are functions and the terms are the outputs of the function.
A recursive formula is a fn rule that relates each term of a sequence after the first to the ones before it.

7, 11, 15, 19 . . .

Find the common difference: FIRST

Write the recursive formula:

Let n = term in sequence

A(n) = the value of the nth term

7, 11, 15, 19 . . .

Let n = term in sequence

A(n) = the value of the nth term

Value of term 1 = A(1) = 7

Value of term 2 = A(2) = A(1) + 4 = 11

Value of term 3 = A(3) = A(2) + 4 = 15

Value of term 4 = A(4) = A(4) + 4 = 19

Value of term 2 = A(2) = A(1) + 4 = 11

Value of term n = A(n) = A(n-2) + 4

The formula for Arithmetic Sequences

A(n) = A(1) + (n – 1) d

Term

number

Common difference

Term number

First term

The formula for Arithmetic Sequences

A(n) = A(1) + (n – 1) d

Example 4: An online auction works as shown: Write a formula for it.

Determine common difference:

A(n) =

What is 12th bid?

The formula for Arithmetic Sequences

Example 5: An RECURSIVE formula is represented by: A(n) = A(n– 1) + 12

If the first term is 19, write explicit formula

So A(1) = 19 and adding 12 is common difference

A(n) = A(1) + (n – 1)d so substitute what you know

A(n) = 19 + (n-1)12 Arithmetic formula

The formula for Arithmetic Sequences

Example 6: An Arithmetic formula is represented by: A(n) = 32 + (n-1)22

So the first term is ?

So common difference is?

A(n) = A(n – 1) + d so substitute what you know

A(n) = Recursive formula

HW p. 279 9 – 42 every third