This document provides information and examples about number patterns, sequence notation, recursive and explicit definitions of sequences, arithmetic sequences, and formulas for finding terms in arithmetic sequences. It includes examples of identifying arithmetic sequences, finding common differences, using recursive and explicit formulas to find specific terms, and using the arithmetic mean to find missing terms. Students are assigned a worksheet to complete problems related to these concepts but are asked to wait on questions 16-20 until they can be discussed.

1. Homework Questions

2. Number Patterns
 Find the next two terms, state a rule to
describe the pattern.
1. 1, 3, 5, 7, 9…
2. 16, 32, 64…
3. 50, 45, 40, 35…
4. -3, -7, -11, -15…

3. Sequence Notation
 A sequence is an ordered list of
numbers – each number is a term.
 State the first 5 terms:
 an = n
 (plug in 1, 2, 3, 4, 5)
 1, 2, 3, 4, 5

4. More Examples
1. an = 4n
2. an = 2n-3
3. an = |1-n2
|
4. an =
5. an = 3
1
n
−
3
6n

5. Recursive v. Explicit

6. Definition
 Recursive Formula – a sequence is
recursively defined if the first term is
given and there is a method of
determining the nth tem by using the
terms that precede it.
 English – if you can use the term before
it to figure out what comes next
 Ex: {-7, -4, -1, 2, 5, …}


7. Examples of Recursive
 {-9, -4, -2, 0, 2, …}
 {-4, -8, -16, -32, -64, …}
 {6, 11, 16, 21, 26, …}
 {8, 4, 2, 1, …}

8. Definition
 Explicit Formula – a formula that
allows direct computation for any term
for a sequence
 English – you don’t need to term prior in
order to figure out what the nth term is
going to be.
 Ex: {8, 9, 10, 11, 12, …}
 an= n + 7

9. Examples of Explicit
 {-3, 1, 5, 9, …}
 {1, 4, 9, 16, …}
 {7, 9, 11, 13, …}
 {24, 20, 16, 12, …}

10. Arithmetic Sequences

11. Arithmetic Sequences
 In an arithmetic sequence, the
difference between consecutive terms
is constant.
 The difference is called the common
difference.
 To find d: 2nd
term – 1st
term

12. Arithmetic?
1. 2, 4, 8, 16
2. 6, 12, 18
3. 48, 45, 42
4. 2, 5, 7, 12

13. Arithmetic Sequence
Formulas
 Recursive Formula
 an = an-1 + d
 use if you know prior
terms
 Explicit Formula
 an = a1 + (n-1)d
 an = nth term
 a1 = 1st
term
 n = number of terms
 d = common
difference

14. Examples
 Find the 20th
term of each sequence
1. 213, 201, 189, 177…
2. .0023, .0025, .0027…

15. More examples
 Find the 17th
term of the sequence:
3. a16 = 18, d = 5

16. Find the missing term
 Use arithmetic mean = average!
4. 84, _______, 110
5. 24, _______, 57

17. Homework
 WORKSHEET!
 We need to talk about numbers 16-20
though, so wait on me!