1. I can identify and extend patterns in sequences and represent sequences using function notation.
4.7 Arithmetic Sequences

2. Sequences
Ordered list of numbers that often form a pattern.
Each number in that list a term of a sequence.

3. Extending a sequence
Find the next 2 terms:
5, 8, 11, 14, …
2.5, 5, 10, 20, …

4. Arithmetic Sequence
The difference between consecutive terms is constant.
The difference is called a common difference.

5. Identifying Arithmetic sequences
Tell whether the following sequences are arithmetic:
3, 8, 13, 18, …
6, 9, 13, 17, …

6. Recursive formula
A function rule that relates each term of a sequence to the term before it.
With exception to the first term
EX: 7, 11, 15, 19, …
The first term, A(1) = 7
The common difference is 4
Next, A(2) = A(1) + 4 = = 11
A(3) = A(2) + 4 = = 15
A(4) = A(3) + 4 = = 19
So, A(n) = A(n – 1) + 4; A(1) = 7 is the recursive formula for this sequence.

7. General formula
A general rule for writing a recursive formula for an arithmetic sequence is:
A(n) = A(n – 1) + d
n represents the term number
d represents the common difference
You must also include the starting value, A(1)

8. You try!
Find the 9th term using a recursive formula:
3, 9, 15, 21, …
A(n) = A(n – 1) + 6; A(1) = 3
A(5) = A(4) + 6 = = 27
A(6) = A(5) + 6 = = 33
A(7) = A(6) + 6 = = 39
A(8) = A(7) + 6 = = 45
A(9) = A(8) + 6 = = 51

9. Practice Write a recursive rule for the following: 70, 77, 84, 91, …
A(n) = A(n – 1) + 7; A(1) = 70
In order to find the 8th term, you need to extend the pattern
To find A(8) I need to know A(7)
To find A(7) I need to know A(6) and so on…
I cannot simply plug in a term number into a recursive formula.

10. For an arithmetic sequence, the formula is:
Explicit formula
A function rule that relates each term of a sequence to the term number.
You can plug it in and find a specific term.
For an arithmetic sequence, the formula is:

11. Try it!
Write an explicit formula:
A(n) = (n – 1)(10)
Find the value of the 12th term
A(12) = (12 – 1)(10)
A(12) = (11)(10) = = 310

12. Explicit from recursive
Here is a recursive formula:
A(n) = A(n – 1) + 12; A(1) = 19
How can I make it explicit?
I know A(1) = 19 and d = 12
A(n) = 19 + (n – 1)(12)

13. Recursive from explicit
Here is an explicit formula:
A(n) = 32 + (n – 1)(22)
What do we know?
A(1) = 32 and d = 22
So, A(n) = A(n – 1) + 22; A(1) = 32

14. Assignment
ODDS
P.279 #9-49