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Today in Precalculus
• Notes: Sequences
• Homework
• Go over quiz
Vocabulary and notation
• Sequences: an ordered progression of
numbers.
• Term: each number in a sequence is a term
First term is a1
Second term is a2
nth term is an
The subscripts denote only the position of
the term in the sequence.
Types
• Arithmetic Sequence: a sequence in which there is a
common difference, d, between every pair of
successive terms.
Example: 5,8,11,14
• Geometric: a sequence in which there is a common
ratio, r, between every pair of successive terms.
Example: 1 , 1 , 1 , 1 ,....
2 4 8 16
Types
• Infinite: there is an infinite number of terms in
the sequence 1 1 1 1
, , , ,....
Example:
2 4 8 16
• Finite: a finite number of terms in the sequence.
Example: 5,8,11,14
• Sequences are infinite unless otherwise
specified.
Explicitly Defined Sequence
• A formula is given for any term in the sequence
Example: ak = 2k - 5
Find the first 5 terms and the 20th term for the
sequence
a1 = 2(1) – 5 = – 3
a2 = 2(2) – 5 = – 1
a3 = 2(3) – 5 = 1
a4 = 2(4) – 5 = 3
a5 = 2(5) – 5 = 5
a20 = 2(20) – 5 = 35
Recursively Defined Sequence
• The first term is given and along with a rule to
obtain each succeeding term from the one
preceding it.
Example: b1 = 8 and bn = bn-1 – 2 for all n>1
Find the next 4 terms for the sequence
b2 = b 1 – 2 = 8 – 2 = 6
b3 = 6 – 2 = 4
b4 = 4 – 2 = 2
b5 = 2 – 2 = 0
General formulas for finding
terms in a sequence
• Arithmetic: an = a1 + (n – 1)d
• Geometric: an = a1r(n–1)
• To use these:
1) Determine if the sequence is arithmetic or
geometric
2) Find the common difference or ratio
Example 1
• Find the 20th term of the sequence 55,49,43, …
and write a recursive and explicit rule.
• Arithmetic sequence with d= -6
• a20 = 55 + (20 – 1)(-6)
a20 = –59
• Recursive rule: ak = ak-1 – 6
• Explicit rule: an = 55 + (n – 1)(-6)
an = 55 – 6n + 6
an = 61 – 6n
Example 2
• Find the 8th term of the sequence
write a recursive and explicit rule.
• Geometric sequence with r=4

1 81
a8  4
5

• Recursive rule: ak = 4ak-1
• Explicit rule: a  1 4n 1
n

5

1 4 16
, , ,... and
5 5 5
16,384
a8 
5
 
1 n
an  (4 ) 4 1
5
1 n
an 
(4 )
20
Practice
Given the sequence 4, -8, 16, -32,…. Determine the
type of sequence, state the recursive and explicit rules
and state the 13th term.
Geometric
Explicit : an = 4 ( -2)
n-1
Recursive :an = -2an-1
a13 = 4(-2) = 16,384
12
Practice
Given the sequence 123, 96, 69,…. Determine the type
of sequence, state the recursive and explicit rules and
state the 13th term.
Arithmetic
Explicit : an = 150 - 27n
Recursive :an = an-1 - 27
a13 = 150 - 13(27) = -201
Homework
• Pg 739: 1-9odd, 21-31odd