# 19-1 and 20-1: Sequences

```19-1 and 20-1: Sequences
Objectives:
1. To represent
sequences using
recursive notation
Assignment:
β’ P. 305: 1-11
β’ P. 319: 1-11
β’ 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, β¦
This is called the Fibonacci
Sequence, and in order to
find the πth 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, β¦
π
Terms of the sequence
1
2
3
4
5
6
ππ 1
1
2
3
5
8
Example 2
Find the next two terms in each of the
following sequences:
1. 7, 12, 17, 22, β¦
2. 7, 35, 175, 875, β¦
Example 2
Find the next two terms in each of the
following sequences:
Arithmetic Sequence
1. 7, 12, 17, 22, β¦
π = common difference
π1 = first term
ππβ1 = term before
the πth term
ππ = πth term
Constant
difference between
consecutive terms
ππ+1 = term after
the πth term
Example 2
Find the next two terms in each of the
following sequences:
Geometric Sequence
2. 7, 35, 175, 875, β¦
π = common ratio
π1 = first term
ππβ1 = term before
the πth term
ππ = πth term
Constant ratio
between
consecutive terms
ππ+1 = term after
the πth term
Example 3
Let ππ be the πth term of the Fibonacci
sequence, where π1 = 1 and π2 = 1. Write a
recursive formula for ππ .
Formulas
A recursive formula
is an equation that
computes a term of a
sequence based on
the term or terms that
precede it.
ππ = ππβ1 + ππβ2
ππ =
1+ 5
π
β 1β 5
π
2π 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.
ππ = ππβ1 + π
ππ = π1 + π β 1 β π
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
π1 = 6 and π = 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.
ππ = π β ππβ1
ππ = π πβ1 β π1
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, β¦
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:
β’ P. 305: 1-11
β’ P. 319: 1-11
β’ Challenge Problems
```
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