* Write the terms of a sequence defined by an explicit formula. * Write the terms of a sequence defined by a recursive formula. * Use factorial notation.

1. 9.1 Sequences and Notations
Chapter 9 Sequences, Probability, and Counting
Theory

2. Concepts & Objectives
⚫ The objectives for this section are
⚫ Write the terms of a sequence defined by an explicit
formula.
⚫ Write the terms of a sequence defined by a recursive
formula.
⚫ Use factorial notation

3. Sequences
⚫ A sequence is a function whose domain is the set of
natural numbers () (the term numbers), and whose
range is the set of term values.
Examples: Find a next term for the following:
A. 3, 10, 17, 24, 31, …
B. 3, 6, 12, 24, 48, …

4. Sequences
⚫ A sequence is a function whose domain is the set of
natural numbers () (the term numbers), and whose
range is the set of term values.
Examples: Find a next term for the following:
A. 3, 10, 17, 24, 31, …
38 (add 7)
B. 3, 6, 12, 24, 48, …
96 (multiply by 2)

5. Sequences (cont.)
⚫ Instead of using f(x) notation to indicate a sequence, it is
customary to use an, where . The letter n is
used instead of x as a reminder that n represents a
natural (counting) number. This is called an explicit
formula.
⚫ The elements in the range of a sequence, called the
terms of the sequence, are . The first term
is found by letting n = 1, the second term is found by
letting n = 2, and so on. The general term, or the nth
term, of the sequence is an.
( )
n
a f n
=
1 2 3
, , , ...
a a a

6. Sequences (cont.)
⚫ You can use Desmos to list the term in a sequence:
⚫ Type the sequence function into Desmos as a
function, f(n).
⚫ Add a table.
⚫ Change the x1 to n1 and y1 to f(n1).
⚫ Enter 1 for n1. When you hit the Enter key, it will fill
in the value for f(n1). Enter 2, and press the Enter key
again, and it will start to populate the list for you.

7. Sequences (cont.)
⚫ Example: Write the first five terms of the sequence.
Step 1: Enter the sequence into Desmos as a function.
1
2
n
n
a
n
+
=
+
(Notice that I
used parentheses
so that Desmos
would divide the
right expression.)

8. Sequences (cont.)
⚫ Example: Write the first five terms of the sequence.
Step 2: Add a table by clicking on the “+” button.
1
2
n
n
a
n
+
=
+

9. Sequences (cont.)
⚫ Example: Write the first five terms of the sequence.
Step 3: Change the x and y.
1
2
n
n
a
n
+
=
+

10. Sequences (cont.)
⚫ Example: Write the first five terms of the sequence.
Step 4: Enter 1-5 for n.
1
2
n
n
a
n
+
=
+
There’re our answers:
a1 = 0.67
a2 = 0.75
a3 = 0.8
a4 = 0.83
a5 = 0.86

11. Piecewise Explicit Formulas
⚫ Generally, sequences are functions whose domain is
over the positive integers. This is true for other types of
functions as well, including some piecewise functions.
(Recall that a piecewise function is a function defined by
multiple subsections.)
⚫ Example: Write the first six terms of the sequence.


= 


2
if is not divisible by 3
if is divisible by 3
3
n
n n
a n
n

12. Piecewise Explicit Formulas
⚫ Example: Write the first six terms of the sequence:


= 


2
if is not divisible by 3
if is divisible by 3
3
n
n n
a n
n
n = 1 1 is not divisible by 3 a1 = 12 = 1
n = 2 2 is not divisible by 3 a2 = 22 = 4
n = 3 3 is divisible by 3 a3 = 33 = 1
n = 4 4 is not divisible by 3 a4 = 42 = 16
n = 5 5 is not divisible by 3 a5 = 52 = 25
n = 6 6 is divisible by 3 a6 = 63 = 2
 
1, 4, 1, 16, 25, 2

13. Writing an Explicit Formula
⚫ Thus far, we have been given the explicit formula and
asked to find a number of terms of the sequence.
Sometimes, the explicit formula for the nth term of a
sequence is not given, and instead, we are given several
terms from the sequence.
⚫ When this happens, we can work in reverse to find an
explicit formula from the first few terms of a sequence.
The key to finding an explicit formula is to look for a
pattern in the terms.

14. Writing an Explicit Formula
⚫ Example: Write an explicit formula for the nth term.
a)
2 3 4 5 6
, , , , ,
11 13 15 17 19
 
− − −
 
 

15. Writing an Explicit Formula
⚫ Example: Write an explicit formula for the nth term.
a)
The terms alternate between positive and negative, so
we can use (‒1)n to make the terms alternate. The
numerator can be represented by n+1.
The denominator is a little trickier, since we need them
to start with 11 and add 2 each time.
2 3 4 5 6
, , , , ,
11 13 15 17 19
 
− − −
 
 

16. Writing an Explicit Formula
⚫ Example: Write an explicit formula for the nth term.
a)
The denominator is a little trickier, since we need it to
start with 11 and add 2 each time.
So, our formula is
2 3 4 5 6
, , , , ,
11 13 15 17 19
 
− − −
 
 
( )
2 1 ? 11 2 9
n
+ =  +
( ) ( )
1 1
2 9
n
n
n
a
n
− +
=
+

17. Writing an Explicit Formula
⚫ Example: Write an explicit formula for the nth term.
b)
2 2 2 2 2
, , , , ,
25 125 625 3,125 15,625
 
− − − − −
 
 

18. Writing an Explicit Formula
⚫ Example: Write an explicit formula for the nth term.
b)
Notice that the terms are all negative and the
numerator is 2.
We can re-write the denominators as power of 5:
2 2 2 2 2
, , , , ,
25 125 625 3,125 15,625
 
− − − − −
 
 
2 3 4 5 6
2 2 2 2 2
, , , , ,
5 5 5 5 5
 
− − − − −
 
 

19. Writing an Explicit Formula
⚫ Example: Write an explicit formula for the nth term.
b)
We can re-write the denominators as power of 5:
So, our formula is
2 2 2 2 2
, , , , ,
25 125 625 3,125 15,625
 
− − − − −
 
 
2 3 4 5 6
2 2 2 2 2
, , , , ,
5 5 5 5 5
 
− − − − −
 
 
1
2
5
n n
a +
= −

20. Recursive Formulas
⚫ Some formulas cannot easily be written using an explicit
formula, but instead depend on the previous terms. The
Fibonacci sequence is an example of this, where the term
is the sum of the previous two terms.
⚫ A formula that defines the terms of a sequence using
previous terms is called a recursive sequence.
⚫ A recursive formula always has two parts: the value of
an initial term(s), and an equation defining an in terms of
preceding terms.

21. Recursive Formulas (cont.)
⚫ Example: Suppose we know the following:
We can find the subsequent terms of the sequence
using the first term.
1
1
3
2 1 for 2
n n
a
a a n
−
=
= − 
a1 = 3
a2 = 2a1 ‒ 1 = 2(3) ‒ 1 = 5
a3 = 2a2 ‒ 1 = 2(5) ‒ 1 = 9
a4 = 2a3 ‒ 1 = 2(9) ‒ 1 = 17  
3, 5, 9, 17

22. Factorial Notation
⚫ An example of a recursive sequence is the product of
consecutive positive integers, called a factorial. n
factorial, written as n!, is the product of the positive
integers from 1 to n.
⚫ For example,
⚫ This is formally written as
⚫ (0! is a special case, and is defined to be 1)
4! 4 3 2 1 24
=    =
( )( ) ( )( )
0! 1
1! 1
! 1 2 2 1 , for 2
n n n n n
=
=
= − − 

23. Factorial Notation (cont.)
⚫ Example: Write the first five terms of
( )
5
2 !
n
n
a
n
=
+
n = 1
n = 2
n = 3
n = 4
n = 5
( )
( )
1
5 1 5 5 5
1 2 ! 3! 3 2 1 6
a = = = =
+  
( )
( )
2
5 2 10 10 5
2 2 ! 4! 4 3 2 1 12
a = = = =
+   
( )
( )
3
5 3 15 15 1
3 2 ! 5! 5 4 3 2 1 8
a = = = =
+    
( )
( )
4
5 4 20 20 1
4 2 ! 6! 6 5 4 3 2 1 36
a = = = =
+     
( )
( )
5
5 5 25 25 5
5 2 ! 7! 7 6 5 4 3 2 1 1008
a = = = =
+      
5
5

24. Factorial Notation (cont.)
⚫ Factorials get large very quickly – faster than even
exponential functions. Depending on the function, the
output may get too large for the calculator.

25. Classwork
⚫ College Algebra 2e
⚫ 9.1: 6-20 (even); 8.4: 26-36 (even); 8.3: 46-54 (even)
⚫ Quiz 8.4