19-1 and 20-1: Sequences
Objectives:
1. To represent
sequences using
recursive notation
Assignment:
• P. 305: 1-11
• P. 319: 1-11
• Challenge Problems
Objective 1
YOU WILL BE ABLE
TO REPRESENT
SEQUENCES USING
RECURSIVE NOTATION
Example 1
Find the first 13 terms in the following
sequence:
1, 1, 2, 3, 5, 8, …
This is called the Fibonacci
Sequence, and in order to
find the 𝑛th term, you need
to know the two that came
before it.
This is idea
behind
recursion.
Sequences
A sequence can be
expressed as a
function whose
domain is the set of
natural numbers.
A sequence is
simply an ordered
list of numbers.
1, 1, 2, 3, 5, 8, …
𝑛
Terms of the sequence
1
2
3
4
5
6
𝒇𝒏 1
1
2
3
5
8
Example 2
Find the next two terms in each of the
following sequences:
1. 7, 12, 17, 22, …
2. 7, 35, 175, 875, …
Example 2
Find the next two terms in each of the
following sequences:
Arithmetic Sequence
1. 7, 12, 17, 22, …
𝑑 = common difference
𝑎1 = first term
𝑎𝑛−1 = term before
the 𝑛th term
𝑎𝑛 = 𝑛th term
Constant
difference between
consecutive terms
𝑎𝑛+1 = term after
the 𝑛th term
Example 2
Find the next two terms in each of the
following sequences:
Geometric Sequence
2. 7, 35, 175, 875, …
𝑟 = common ratio
𝑎1 = first term
𝑎𝑛−1 = term before
the 𝑛th term
𝑎𝑛 = 𝑛th term
Constant ratio
between
consecutive terms
𝑎𝑛+1 = term after
the 𝑛th term
Example 3
Let 𝑓𝑛 be the 𝑛th term of the Fibonacci
sequence, where 𝑓1 = 1 and 𝑓2 = 1. Write a
recursive formula for 𝑓𝑛 .
Formulas
A recursive formula
is an equation that
computes a term of a
sequence based on
the term or terms that
precede it.
𝑓𝑛 = 𝑓𝑛−1 + 𝑓𝑛−2
𝑓𝑛 =
1+ 5
𝑛
− 1− 5
𝑛
2𝑛 5
An explicit formula
is an equation that
computes any term
of a sequence
directly.
Arithmetic Formulas
A recursive formula
is an equation that
computes a term of a
sequence based on
the term or terms that
precede it.
𝑎𝑛 = 𝑎𝑛−1 + 𝑑
𝑎𝑛 = 𝑎1 + 𝑛 − 1 ∙ 𝑑
An explicit formula
is an equation that
computes any term
of a sequence
directly.
Example 4
Write a recursive formula and an explicit
formula for the arithmetic sequence with
𝑎1 = 6 and 𝑑 = 3. Then find the 40th term of
the sequence.
Geometric Formulas
A recursive formula
is an equation that
computes a term of a
sequence based on
the term or terms that
precede it.
𝑎𝑛 = 𝑟 ∙ 𝑎𝑛−1
𝑎𝑛 = 𝑟 𝑛−1 ∙ 𝑎1
An explicit formula
is an equation that
computes any term
of a sequence
directly.
Example 5
Write a recursive formula and an explicit
formula for the sequence below. Then find
the 40th term of the sequence.
4096, 1024, 256, 64, …
Example 6
1. What type of function
is a geometric
sequence?
2. What type of function
is an arithmetic
sequence?
19-1 and 20-1: Sequences
Objectives:
1. To represent
sequences using
recursive notation
Assignment:
• P. 305: 1-11
• P. 319: 1-11
• Challenge Problems