Typically, only the first 3 or 4 terms of a
sequence are listed, followed by an ellipsis (three periods which
mean "and so on"). The following table shows several
numerical sequences of the counting numbers (1,2,3,4,...)
Name of Sequence
The key to continuing a sequence is looking
for a pattern. What gets us from one term to the next?
Numerical sequences can be classified according to the method
used to get from one term to the next. For example, if we add a constant
as we move from one term to the next, the sequence is called an arithmetic
sequence (in the arithmetic sequence in the table, we add 5 to each term
to get the next term). If we multiply by a constant as we move from
one term to the next, the sequence is called a geometric sequence (in
the geometric sequence in the table, we multiply each term by 2 to get the
We can also look for patterns in sequences
of geometric figures. Consider the following figures. Can you
identify the pattern and extend the sequence by giving the next
Check for Understanding
Choose each answer from the
Determine whether the following sequence (the number of dots
in each triangular array) is arithmetic, geometric, or neither.
External Links to Extend Basic Understanding
Harcourt's Interactive Math Glossary: Read another definition
sequence, and geometric
Related Investigations on InterMath to
Completing the Sequence: Find the
next numbers in the sequences.
Triangular Numbers: Form triangular
numbers and look for patterns.
Spreading Rumors: It is important for 8000 people to hear a rumor by May 14. Is this
outcome likely to occur if the pattern for spreading the rumor continues as
Cutting the Cake: Cut a square cake into n pieces so that
all the pieces have equal amounts of cake and equal amounts of frosting.
The 26th Degree: Simplify an expression.
A series is a summation of numbers in which
the terms being added follow a particular pattern. An example
of a series is:
in which the number in the denominator increases
Series can end (be finite) or continue forever
(be infinite, indicated by the "..." at the end. The
above series is an infinite series. An example of a finite series
in which each successive term is multiplied
by x. This particular series is known as a geometric series -
a series in which each term is a constant multiple of the term
Enter your answer and press tab
after each response
Going Fishing: Determine the terms of an arithmetic sequence given
its constant difference, number of terms, and sum.
Infinite Series: Determine the sum
of various infinite series, and how you can tell if a series will
Hot Air Balloons: Determine the height of a
hot air balloon over time based on a recursive condition.
Sum of Natural Numbers: Find a
way to sum a set of natural numbers.
Half-Sums: Consider the infinite sums of numbers.
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Examples of Arithmetic Sequence
Consider the following arithmetic sequences of
numbers. Could you have figured out what the common difference is in each
(Fixed Number Being Added)
2, 3.3, 4.6, 5.9, 7.2,
1, 3, 5, 7, ...
As indicated in the second example, the difference between any
two consecutive terms does not have to be a whole number! In this
case, it is 1.3.
The sequence of odd numbers (the third example) forms an
arithmetic sequence since the common difference is 2. Which other numerical
sequences of the counting numbers (see Table 1) form arithmetic sequences?
Interactive Math Glossary: Read another definition of arithmetic sequence.
Pick up the Sticks: Devise
a strategy so that you're the one leaving the last stick on the
table every time.
Average Groups: Determine the arithmetic mean of a group of terms in a sequence.
Growing Tree: After four years, how many feet high was a tree that Mrs. Johnson's
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Examples of Geometric Sequences
Consider the following geometric sequences of numbers.
Could you have figured out what the common ratio is in each sequence?
(Fixed Number Being Multiplied)
As indicated in the second example, the constant
factor does not have to be a positive number or even a whole number!
In this case, it is -0.5.
Interactive Math Glossary: Read another defintion of geometric sequence.
Geometric Sequences and
Series: Explore the properties of geometric sequences and
Bouncing Ball: Determine how high a ball rebounds with each bounce.
Double Dollars: What is the fewest number of years until
an investment doubles in value?
Bouncing Ball II: Determine how far a golf ball will rebound.
This sequence of numbers was discovered by an Italian mathematician,
Leonardo of Pisa (nicknamed Fibonacci), in 1202 when he was investigating
how fast rabbits can breed in ideal conditions. Click
here to read more about this discovery as well as extensions, like
Dudeney's Cows and bees, Fibonaaci numbers, and family trees!
Where Can We See Fibonacci
Well, Don't Mess With Mother Nature!
We can see Fibonacci numbers in nature. For example, consider
the sunflower. The head of a sunflower is formed into two opposite sets of
spirals, 21 spirals going in one direction and 34 going in the other. These
are consecutive numbers in the Fibonacci sequence.
There are other spirals in nature that also
have Fibonacci numbers. The pine cone has 5 spirals going in one
direction and 8 spirals going in the other. The pineapple has
8 and 13.
We can even see the Fibonacci sequence in tree
branch growth. Start at the bottom and count the number of branches
cut by the lines. You will get 1, 2, 3, 5, ...
Is all this just a coincidence? I don't know,
but remember, "Don't mess with Mother Nature!"
More on the Fibonacci
Interactive Math Glossary: Read another definition of the Fibonacci
Multiplying Rabbits: Examine how the family tree of a rabbit forms the Fibonacci sequence
and the golden ratio.
Fibonacci Extended: Identify an explicit
pattern between terms in a group of numbers that are generated
Ratio of Fibonacci: Use the Fibonacci sequence
to examine ratios.
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