Unit 3 Linear and Exponential
Functions
Lesson
Linear and Exponential Number Patterns
Sequences
 A set of numbers written in a specific order
or pattern
 Examples
 2, 4, 6, 8 -- finite sequence (only 4 terms)
 0, 5, 10, 15, … -- infinite sequence (infinite
# of terms)
Terminology/Notation
 Term – a number in a sequence
 a1 -- represents the first term in a
sequence (1 is the counter)
 an -- the nth term in a sequence or the
formula
an 1 -- the term before the nth term
Sequence Formulas
 Ex.
2
 For the sequence formula, an  n  5 ,
 solve the first 5 terms.
Consider the series 4, 8, 12, 16, 20, 24, …
The third term in the series, a3, is 12.
What is the first term?
What is the sixth term?
What is the seventh term?
In the previous example, we came up with
the seventh term by looking at the pattern
and applying it to the next term in the
sequence.
But what if we are asked to find the 150th
term?
Today, we are going to look at 2 ways to
write a rule (an equation) for finding the nth
term in a series: closed formula and
recursive formula.
Closed or Explicit Formula
With a closed formula, we do not need to
know what the previous terms are in
order to calculate the next term.
Let’s practice:
Find the 150th term: an = n2 – 4
Linear Sequences
You’ve already been using “closed formula”
sequences, for example linear functions:
y= 2x - 2 graphs the linear function, an = 2n - 2
graphs linear sequence dots.
 Ex.
 an = 2n – 2
 Chart
Graph
Linear Sequences = Arithmetic Sequences
 For linear sequences, you add the same
amount each time
 Arithmetic Sequence Explicit Formula

an  mn  b
or
an  a1   n 1 d
1, 3, 5, 7, 9, …
The closed formula involves a linear function with
slope of _________ and a y-intercept (0 term) of
___________.
Answer:
an  2n  1
Let’s look at another example:
4, 8, 12, 16, 20, …
The slope is _______________.
The y-intercept is _____________.
Answer: an  4n
Example
 Write the explicit formula for:
 -22, -2, 18, 38 …
Example
 Solve the explicit formula for an arithmetic
sequence with a20  209 and d = -10.
Example
 Solve the explicit formula if
a19  3597 and
a40  7797
1, 3, 9, 27, 81, …
What is the sequence pattern?
Chart
Graph
1, 3, 9, 27, 81, …
Exponential Closed Sequence
We can say the exponential function’s ratio (or
base) is ______.
The function’s 1st term or
Answer:
n1
an  1(3)
a1
is _________.
Exponential Sequences =
Geometric Sequences
an  a1r
n 1
 If r is between 0 and 1 --- decays
 If r > 1 --- grows
Recursive Formula
What does recursive mean?
The dictionary defines recursive as
pertaining to or using a rule or procedure
that can be applied repeatedly.
So, to simplify things, when you see
recursive, I want you to think repeat.
Example of a recursive formula
a1 = 4
an = an-1 + 4
Notice that the formula has 2 parts:
1. It defines the first term. In the above example,
a1 = 4.
2. It defines the remaining terms. In our
example, an=an-1 + 4.
What does an-1 mean???
Find the first six terms of the
recursive sequence
a1 = 3
an = an-1 + 5
a1 = 3
a2 = 3 + 5 = 8
a3 = 8 + 5 = 13
a4 = 13 + 5 = 18
a5 = 18 + 5 = 24
a6 = 24 + 5 = 29
Repeat
Repeat
Repeat
Repeat
If you wanted to find the 50th term, you
would have to repeat your calculations 50
times.
Can you see why we associate recursive
with repeat?
Just to summarize
A closed formula has only one equation. You can
easily find any term in the sequence.
*****
A recursive formula has 2 parts: the first term, and
a rule, or equation, for finding the remaining
terms based on knowing the previous term.
In order to find the 50th term, you will first have to
find the previous 49 terms.
1, 3, 5, 7, 9, …
In the series above, we notice that each term in
the series is just 2 more than the term before it.
We can say that
a1 = 1
a2 = a 1 + 2
a3 = a 2 + 2
a4 = a 3 + 2
Instead of writing a2, a3,
a4, etc, we can shorten it
to: an = an-1 + 2
1, 3, 5, 7, 9, …
So our recursive formula is:
a1 = 1
an = an-1 + 2
Notice our formula has both parts: a1 and an
You try…
Write the closed linear function and the
recursive formula for the following
sequences:
1) 0, 5, 10, 15, 20, …
2) 3, 10, 17, 24, …
3) 1, 11, 21, 31, …
4) 2, 4, 6, 8, …
Let’s write the closed formula for
the four sequences we just defined.
Pay attention to the pattern…
Problem 1:
0, 5, 10, 15, 20, …
Problem 2:
3, 10, 17, 24, …
Recursive:
a1 = 0
an = an-1 + 5
Recursive:
a1 = 3
an = an-1 + 7
Closed:
an = 5n - 5
Closed:
an = 7n - 4
Can you see the pattern???
Problem 3:
1, 11, 21, 31, …
Problem 4:
2, 4, 6, 8, …
Recursive:
a1 = 1
an = an-1 + 10
Recursive:
a1 = 2
an = an-1 + 2
Closed:
an = 10(n-1) + 1
Closed:
an = 2(n-1) + 2
1, 3, 9, 27, 81, …
In the series above, we notice that each term in
the series is 3 times the term before it.
We can say that
a1 = 1
a2 = 3a1
a3 = 3a2
a4 = 3a3
Instead of writing a2, a3,
a4, etc, we can shorten it
to: an = 3an-1
1, 3, 9, 27, 81, …
So our recursive formula is:
a1 = 1
an = 3an-1
Notice our formula has both parts: a1 and an
You try…
Write the closed and recursive formula for
the following sequences:
1) 1, 5, 25, 125, …
2) 3, 12, 48, 192, …
3) 4, 24, 144, 864, …
4) 2, 20, 200, 2000, …
Let’s write the closed formula for
the four sequences we just defined.
Pay attention to the pattern…
Problem 1:
1, 5, 25, 125, …
Recursive:
a1 = 1
an = 5an-1
Problem 2:
3, 12, 48, 192, …
Recursive:
a1 = 3
an = 4an-1
Closed:
an = 5(n-1)
Check: a4 = 53 = 125
Closed:
an = 3*4n-1
Check: a4 = 3(4)3 = 192
Let’s write the closed formula for
the four sequences we just defined.
Pay attention to the pattern…
Problem 3:
4, 24, 144, 864, …
Recursive:
a1 = 4
an = 6an-1
Problem 4:
2, 20, 200, 2000, …
Recursive:
a1 = 2
an = 10an-1
Closed:
an = 4*6(n-1)
Check: a3 = 4(6)2 = 144
Closed:
an = 2*10n-1
Check: a4 = 2(10)3 = 2000