Arithmetic and Geometric Sequences
ARITHMETIC AND GEOMETRIC SEQUENCES
Geometric and Arithmetic Sequences
 A sequence is a set of numbers, called terms, arranged in a particular order.
 An arithmetic sequence is a sequence with the difference between two consecutive terms
constant. The difference is called the common difference.
 A geometric sequence is a sequence with the ratio between two consecutive constant terms. This
ratio is called the common ratio.
Examples:
Arithmetic Sequence: 2, 4, 6, 8, 10, …
Each term in this sequence equals the term before it with 2 added on. The common difference is 2 and
notice that it is growing at a constant rate. We could also think of this sequence as the output from a
function, f(n) = 2n, where the domain is restricted to positive integers.
Explicit
formula:
Recursive
formula:
an  2n
a1  2
(says: for the new
number “a” at “n”
stage”, multiply
the stage by 2)
(says: the first number
is 2)
(or think of it as
the rule for the nth
term)
an  an1  2
(says: to get the new
number “a”, take the
number before and add
2)
Certain sequences, such as
this arithmetic sequence,
can be represented in more
than one manner. This
sequence can be
represented as either an
explicit formula (an = a1 +
(n – 1)d) or a recursive
formula (an = an – 1 + d).
Graphical representation:
f(n) = 2n
Geometric sequence: 3, 9, 27, 81, ...
Each term is equal to the prior one multiplied by 3. That means the common ratio is 3. With this sequence
the rate of change is not constant. It is increasing with each term, but each increase is proportional to the
previous term. Again, this sequence could be considered as the output from a function, g(n) = 3n with the
domain restricted to positive integers. Geometric sequences grow more quickly than arithmetic sequences.
Explicit
formula:
Recursive
formula:
an  3n
a1  3
(says: for the new
number “a” at “n”
stage”, multiply by
3 “n” times))
(says: the first
number is 3)
(or think of it as the
rule for the nth term
(says: to get the new
number “a”, multiply
the previous number
by 3)
an  3an1
Certain sequences, such as
this geometric sequence, can
be represented in more than
one manner. This sequence
can be represented as either
an explicit formula (an =
a1 • rn – 1) or a recursive
formula (an = an – 1 • r).
Graphical representation:
f(n) = 3n
Neither Arithmetic or Geometric: 1, 2, 3, 4, 3, 2, 1, 2, 3, …
The terms in this sequence change by 1, but sometimes it is added and sometimes it is subtracted. It is not
an arithmetic sequence. There is no common value being multiplied so the sequence is not geometric.
ARITHMETIC AND GEOMETRIC SEQUENCES
Students should be able to describe an arithmetic or geometric sequence recursively as well as explicitly.
They should be able to explain how the patterns they see in the sequence relate to the explicit expression or
function.
Patterning and sequences are a great way to begin to understand the similarities and differences between
linear and exponential functions.
Examples:
Pattern 1: Suppose this is a pattern created by toothpicks. How many toothpicks are needed to create the
5th step? How many toothpicks on the nth step”? Use pictures, words or symbols to represent how this
pattern is changing.
Step 1
Step 2
Step 3
Step 4
Pattern 2: How many “X” in step 5? How many in step n? Use pictures, words or symbols to represent
how this pattern is changing.
Step 1
Step 2
Step 3
Step 4
xx
x
xxxx
xx
xxxxxxxx
xxxx
xxxxxxxxxxxxxxxx
xxxxxxxx
Both Pattern 1 and Pattern 2 are growing patterns. Let’s explore each of them for a moment without using
a table right away.
For Pattern 1 there are a few ways that you might “see” how the pattern grows:
One way to show that is by writing the sequence of values for how many toothpicks are in each step:
4, 7, 10, 13…
Other ways students may see Pattern 1
4
4
2+2
4+3
4+4–1
4+3
4+3+3
4 + (4 – 1) + (4 – 1)
6+4
4+3+3+3
4 + (4 – 1) + (4 – 1) + (4 – 1)
8+5
…4 + 3(n-1) or 3n + 1
…4 + (n – 1) + (4 – 1) or 3n + 1 …2n + (n + 1) or 3n + 1
You could also write out in words: Start with 4 and then add 3 for each successive step.
This is a verbal expression of the recursive expression:
a0  4
an  an 1  3
ARITHMETIC AND GEOMETRIC SEQUENCES (continued)
The above shows that students may “see” the pattern in a number of ways, but regardless, all valid
explications of the patterns will simplify to the same linear equation. It is important that students “see” the
pattern from different perspectives and be comfortable algebraically representing different perspectives. In
this case, all patterns simplify to:
y = 3x + 1 where y is the total number of toothpicks and x is the step number.
This obviously is an arithmetic sequence and the common difference is a constant rate of change which is
why this pattern can be represented by a linear function.
Now let’s look at Pattern 2.
Ways students may see Pattern 2
3•1
3•2
3•4
3•8
1+2
2+4
4+8
8 + 16
2n – 1 + 2n = 2n – 1(1 + 2) = 2n -1 • 3
= 3 • 2n -1
3 • 2n -1
1 • 3 = 20 • 3
2 • 3 = 21 • 3
2 • 2 • 3 = 22 • 3
2 • 2 • 2 • 3 = 23 • 3
2n -1 • 3 = 3 • 2n -1
Regardless of how the student sees the pattern, the sequence is: 3, 6, 12, 24, …
The pattern is not increasing by a common difference which tells us that it is not an arithmetic sequence.
You could also write out in words: Start with 3 and then multiple by 3 for each successive step.
This is a verbal expression of the recursive expression:
a0  3
an  an 1 2
Again, students will see the pattern differently, but ultimately, all the patterns are the same. In this case, all
patterns simplify to:
y = 3(2x-1) or where y is the total number of “X’s” and x is the step number.
This obviously is a geometric sequence and the common ratio which is why this pattern can be represented
by an exponential function.
Now let’s look at a table of values for both Pattern 1 and Pattern 2.
Pattern 1:
Step
Numbe
r
1
2
3
4
Pattern 2:
Total
Number of
Toothpick
s
4
7
10
13
Differenc
e
Chang
e
Up 3
Up 3
Up 3
+3
+3
+3
Step
Numbe
r
1
2
3
4
Total
Numbe
r of
“X”
3
6
12
24
Differenc
e
Chang
e
Up 3
Up 6
Up 12
×2
×2
×2
The tables help us to see the difference between growth by equal intervals (linear) and growth by equal
factors (exponential.) Notice that Pattern 2 is increasing more rapidly than Pattern 1. There is a
significant difference between linear and exponential growth.
Constructing Linear and Exponential Functions
CONSTRUCTING LINEAR AND EXPONENTIAL FUNCTIONS
These goals are very much connected to other “I can” statements in this unit. The examples below give an
idea of what you could give students and then expect them to be able to write an algebraic expression of the
function.
Examples:
1. For the linear function, f ( x )  3 x  1 , students could be given any of the representations below and
then asked to come up with the function.
Arithmetic sequence: 1, 4, 7, 10, 13, 16, 19, …
Verbal description: An initial quantity is 1 and it increases by a constant rate of 3 each time.
Table of input/output values:
x
0
1
2
3
4
5
6
2.
Graph:
f(x)
1
4
7
10
13
16
19
For the exponential function, g ( x)  2 x , students could be given any of the following representations
in order to construct the function.
Geometric sequence: 1, 2, 4, 8, 16, 32, 64, …
Verbal description: A certain type of bacteria reproduces by the organism dividing into two.
A new bacteria culture is started by putting one cell on a new agar plate.
Table of input/output values:
x
0
1
2
3
4
5
6
g(x)
1
2
4
8
16
32
64
Graph:
Linear and Exponential Functions
LINEAR FUNCTIONS GROW BY EQUAL DIFFERENCES
1
2
a difference of 5. Looking at the interval 20  x  10 , which is the same size as the previous interval, the y value goes from 15
Example: The graph right shows the function y   x  5 . Notice that over the interval 0  x  10 the y value goes from 5 to 0,
to 10, also a difference of 5. This shows that for a linear function, the rate of change is the same over equal intervals. The same
conclusion can be drawn from the difference table below.
x
y
x interval
Difference
in y
-20
-15
-10
-5
0
5
10
15
20
15
12.5
10
7.5
5
2.5
0
-2.5
-5
5
5
5
5
5
5
5
5
- 2.5
- 2.5
- 2.5
- 2.5
- 2.5
- 2.5
- 2.5
- 2.5
5
-20 ≤ x ≤ -10
5
0 ≤ x ≤ 10
EXPONENTIAL FUNCTIONS GROW BY EQUAL FACTORS
Example: The graph at the right shows the function y  3x . Notice that over the interval 0  x  1the y value goes from 1 to 3, a
difference of 2. For the interval 1  x  2 , which is the same size as the previous interval, the y value goes from 3 to 9, a difference
of 6. This shows that for an exponential function, the rate of change is not constant over equal intervals. However, if you take the
ratio of the differences in the y-values you can see that each increase is by a factor of 3. The same conclusion can be drawn from
the difference table below.
x
Difference
in y
y
x interval
1
27
1
9
1
3
1
0
1
1
1
3
1
2
2
3
9
27
1
1
6
18
-3
-2
-1
1
2
27
2
9
2
3
Ratios of
differences
2 2

3
9 27
2 2
 3
3 9
2
2  3
3
62  3
18  6  3
6
2
1≤x≤2
0≤x≤1
MODELING LINEAR AND EXPONENTIAL FUNCTIONS
Students should recognize that situations that can be modeled with a linear function can be identified by having a constant rate of
change, whereas situations where an exponential model is appropriate have a rate of change that is a constant percent rate.
Examples: Identify whether a linear or exponential function should be used to model each of the following situations.
1. James had surgery on his left knee. As part of his rehabilitation, the physical therapist recommends that he start jogging.
James is to jog for 12 minutes each day for the first week. Each week thereafter, James is to increase the time that he jogs
each day by 6 minutes. (Linear)
2. The sum of the interior angles of a triangle is 180º, of a quadrilateral is 360º, of a pentagon is 540º and of a hexagon
is720º. (Linear)
3. A culture of bacteria contains 500 individual organisms and doubles every 2 hours. (Exponential)
A mine worker discovers an ore sample containing 500 mg of a radioactive material. It is discovered that the radioactive material
has a half life of 1 day. (Exponential)
4 Help: Comparing Linear & Exponential Functions
COMPARING LINEAR AND/OR EXPONENTIAL FUNCTIONS
Comparing linear functions:
Graph
Rate of
Change
linear
y = mx + b
slope of m = 0
constant function
y = 0x – 3
therefore, y = -3
slope of
zero
Table
x-int.
x
-3
-2
-1
0
1
2
3
y
-3
-3
-3
-3
-3
-3
-3
x
-3
-2
-1
0
1
2
3
y
-7
-4
-1
2
5
8
11
x
-3
-2
-1
0
1
2
3
y
3.5
3
2.5
2
1.5
1
.5
y-int.
Domain
x values
(0, -3)
none
or
Range
y values
y = -3
all real
numbers
(0, b)
or
y=b
(hence b = -3)
linear
y = mx + b
slope of m > 0
linear increasing
increasing
at a
constant
rate of m
units
y = 3x + 2
next =
now + m
starting at
b
linear
y = mx + b
decreasing
at a
constant
rate of m
units
slope of m < 0
linear decreasing
y = -.5x + 2
next =
now + m
starting at
b
 2 
 ,0
 3 
or
 b 
  ,0
 m 
(0, 2)
or
all real
numbers
all real
numbers
all real
numbers
all real
numbers
(0, b)
(4, 0)
or
 b 
  ,0
 m 
(0, 2)
or
(0, b)
COMPARING LINEAR AND/OR EXPONENTIAL FUNCTIONS
Comparing exponential functions:
Graph
exponential
growth
y = abx
b>1
*growth
factor b
y = 2(3)x
Exponential
decay
y = abx
0<b<1
*decay
factor b
y = 2(.5)x
Rate of
Change
increasing
at an
increasing
rate
next =
now • 3
starting at
2
decreasing
at a
decreasing
rate
next =
now • .5
starting at
2
Table
x
0
1
2
3
4
5
6
y
2
6
18
54
162
486
1458
x
0
1
2
3
4
5
6
y
2
1
.5
.25
.125
.0625
.03125
x-int.
none
x-axis is
an
asympto
te
none
x-axis is
an
asympto
te
y-int.
Domain
x values
Range
y values
all real
numbers
y>0
all real
numbers
y>0
(0, 2)
or
(0, a)
(0, 2)
or
(0, a)
In a linear equation mx represents additive change; (b + m + m + m + m + ……….)
 the slope m is the constant rate of change between any two points on the linear graph/table
y
y  y 

 x1 , y1  and  x2 , y2  = 2 1 , also called
x
 x2  x1 
*In
an exponential equation, bx represents multiplicative change; (a • b • b • b • b • ……….)
 Growth factor b = 100% + growth rate (percent)
Example: Value of savings account is growing by 2.5% per year, the growth fact is 1.025. y = balance (1.025)x
 Decay factor b = 100% – decay rate (percent)
Example: Value of the dollar is declining each year by 3%, the decay factor is 97%. y = 1(.97)x
 The average rate of change of points in an exponential graph/table will not be constant
 The average rate of change between any two points is the slope of the secant line between two points (linear or non-linear).
 y2  y1 
x
2
 x1 
As students explore the similarities and differences between linear and exponential function, use correct terminology for constant
rate of change (slope of a line) and average rate of change (slope of secant line).
COMPARING LINEAR AND/OR EXPONENTIAL FUNCTIONS
Students need to be comfortable working with functions in ALL representations. They should be fluent in moving between the different
representations and should be able to identify characteristics of an exponential function and linear function whether the representation is a
table, an algebraic equation, a graph, a verbal description or a real-world context.
Table
Algebraic
Equations
Graph
Verbal
Description
Context
Example:
Discuss and compare
the following
functions:
3
f ( x)  x  5
2
and
g ( x)  3x  1
g ( x)  3  1
x
Examine the different
representations of the
functions – i.e. tables,
graphs, equations,
etc. Discussion and
comparisons should
include: identifying
differences in rates of
change, intercepts,
and/or where each
function is greater or
less than the other.
3
f ( x)  x  5
2
x
-2
f(x)
2
-1
3.5
0
1
2
5
6.5
8
difference
Rate of change:
1.5
1.5
1.5
1.5
3
2
x
-2
-1
0
1
2
g(x)
difference
1.11
1.33
2
4
10
2
6
Rate of change: not constant
Here we are comparing a linear function, f(x) with an exponential function, g(x). Notice that in the linear equation the variable, x, is
multiplied by the coefficient, but in the exponential equation the x is the exponent. So with the exponential function, the inputs become the
exponent. Both functions have a constant added on the end. In f(x), the constant, 5, tells us where the y-intercept of the graph will end up. In
g(x), the constant, 1, tells us where that the horizontal asymptote of the function will be y = 1. These facts can both be verified by examining
the graphs of both functions.
Looking at the tables for the two functions, we can examine the rates of change for each function. For f(x), the rate of change stays the same
3
– it is . In contrast, g(x) has a rate of change that does not remain constant. The rate of change there is being multiplied by a factor of 3
2
each time. The rates of change are both increasing, but g(x) is increasing at a faster rate than f(x).
Increasing Exponentially and Linearly
INCREASING EXPONENTIALLY AND LINEARLY
Students should be able to use all of the different representations of a function to explain why a quantity
that is increasing exponentially will eventually be much larger than one that is increasing linearly. That is
they should be able to explain it verbally, by
using a graph, a table of values, etc.
Examples:
1. The graph at the right shows a linear
function, y  2 x  5 , in blue, and an
exponential function, y  2 , in red. In
looking at the graphs you can see that the
linear function has higher y-values when
the x-values are in between -2 and 3.
However, when the x-values increase above
4, the exponential function increases more
quickly and has larger y-values than the
linear function.
y = 2x + 5
x
y = 2x
Note: If using technology, students could
find the points of intersection of the two
graphs and give precise intervals when
comparing the two functions.
2.
Suppose a company offers you a choice in how you are paid:
Option A:
You can earn $10,000 a day for 30 days.
Option B:
You earn $1 on the first day, $2 on the second day, $4 on the third day, $8 on the fourth day, and so
forth. In other words, they offer to pay you $1 on the first day and then double your pay each
successive day for 30 days.
Which option is better?
Solution:
Option A is fairly straightforward; each day you earn $10,000. So at the end of 30 days, you will earn
$100,000 x 30 = $300,000.
Option B seems a little less enticing given that on day one you get $1, day two $2, day three $4, and so
forth, it just doesn’t seem like a lot of money. But let’s write out the pattern further:
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768…
In other words, by day 15 you earn more than $10,000 in a day. As a matter of fact, if you were to
continue to double the pay each day, by day 30 the pay would be $535,870,912. Hence, Option B is a
far better option than Option A!
Help Links
Arithmetic Sequences:
 http://www.mathguide.com/lessons/SequenceArithmetic.html
Geometric Sequences:
 http://www.mathguide.com/lessons/SequenceGeometric.html
Arithmetic and Geometric Sequences:
 http://www.mathsisfun.com/algebra/sequences-series.html
 http://www.purplemath.com/modules/series3.htm
 http://tutorial.math.lamar.edu/Classes/CalcII/Sequences.aspx
 http://home.windstream.net/okrebs/page131.html
Recursion and Notation explained:
 http://www.learner.org/workshops/algebra/workshop5/index2.html#3