This document provides information about number patterns, sequences, and series. It defines recursive and explicit formulas for sequences and discusses arithmetic and geometric sequences. For arithmetic sequences, the common difference is constant between terms. For geometric sequences, the common ratio between consecutive terms is constant. Formulas are provided to find later terms in sequences, whether defined recursively or explicitly. Examples are given to practice identifying sequence types and using the formulas to find missing or future terms.

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. Geometric Sequences

18. Geometric Sequences
 In a geometric sequence, the ratio
between consecutive terms is constant.
 This ratio is called the common ratio.
 To find r:
stterm
ndterm
1
2

19. Geometric, Arithmetic, Neither?
(find the next 2 terms if so)
1. 5, 15, 45, 135…
2. 15, 30, 45, 60…
3. 6, -24, 96, -384…
4. 8, 20, 32, 44…

20. Geometric Sequences Formulas
 Recursive Formula
 an = an-1 r
 Explicit Formula
 an = a1 rn-1
•
•

21. Find the 19th
term…
1. 11, 33, 99, 297…
2. 20, 17, 14, 11, 8…

22. FYI - Graphs
 Arithmetic Graphs are linear
 Geometric Graphs are exponential

23. Geometric Mean
 Geometric Mean =
3. 20, _____, 80
4. 3, ____, 18, 75
5. 28, ____, 5103
numbersproductof 2

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