InterMath Dictionary

Sequences

 Arithmetic Sequence  Geometric Sequence
 Fibonacci Sequence  Sequence
 Series


Sequence - A pattern involving an ordered arrangement of numbers, geometric figures, letters, or other objects. The individual numbers, figures, or letters in the sequence are called the terms of the sequence.
Series - A summation of the terms in a sequence.

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,...)

 Sequence

 Name of Sequence

 2, 4, 6, 8, ...  the even (counting/natural) numbers
 1, 3, 5, 7, ...  the odd (counting/natural) numbers
 1, 4, 9, 16, ...  the square (counting/natural) numbers
   the powers of 7
 1, 1, 2, 3, 5, 8, ...  the Fibonacci sequence
 5, 10, 15, ...  an arithmetic sequence
 5, 10, 20, 40, ...  a geometric sequence

Table 1

 

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 next term.).

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 3 terms?

 

Check for Understanding

Determine if the following sequences are arithmetic, geometric, or neither.

 Choose each answer from the menus.

 
 a) 1, 5, 9, 13, ...
a)
 b) 3, 4, 7, 11, ...
b)
 c) -2, 6, -18, 54, ...
c)
 d) 5, 2, -1, -4, ...
d)
 e) 1/8, 1/2, 2, 8, ...
e)

Determine whether the following sequence (the number of dots in each triangular array) is arithmetic, geometric, or neither.

f)

 

Check for Understanding

Determine the missing value in each of the following sequences. Enter your answer and press tab after each response.

Graded Response

 g) 3, 6, , 24, ...
g)
 h) , -9, -3, 3, ...
h)
 i) 1, 4, 9,
i)

External Links to Extend Basic Understanding

Harcourt's Interactive Math Glossary: Read another definition of sequence, arithmetic sequence, and geometric sequence.

Related Investigations on InterMath to Challenge Teachers

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 started?
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.


Series

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 by 1.

 

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 is:

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 before.

 

Check for Understanding

Consider the series -5 + 0 + 5 + ... + 35

Enter your answer and press tab after each response

Graded Response

 a) How many terms are in the series ?
a)
 b) What is the sum of the series ?
b)

 

Related Investigations on InterMath to Challenge Teachers

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 converge.
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.

Top of Page


Arithmetic Sequences

Arithmetic Sequence - A sequence of numbers in which the difference between any two consecutive terms is the same. In other words, you add the same number each time to get the next term in the sequence. This fixed number is called the common difference for the sequence.

Examples of Arithmetic Sequence
Consider the following arithmetic sequences of numbers. Could you have figured out what the common difference is in each sequence?

 Sequence

Common Difference 
(
Fixed Number Being Added)

 3,6,9,12,15,...

 3

 2, 3.3, 4.6, 5.9, 7.2, 8.5, ...

 1.3

 1, 3, 5, 7, ...

 2


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?

 

Check for Understanding

Answer the following questions about arithmetic sequences.

Enter your answer and press tab after each response

Graded Response

 a) Find the common difference in the sequence
4, 13, 22, 31, ...
a)
 b) The first term in an arithmetic sequence is 8 and the eight term is 36. How far apart are each of the terms in the sequence ?
b)

 

External Links to Extend Basic Understanding

Harcourt's Interactive Math Glossary: Read another definition of arithmetic sequence.

Related Investigations on InterMath to Challenge Teachers

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 class planted?

Top of Page



Geometric Sequences
Geometric Sequence - A sequence of numbers in which the ratio between any two consecutive terms is the same. In other words, you multiply by the same number each time to get the next term in the sequence. This fixed number is called the common ratio for the sequence.

Examples of Geometric Sequences
Consider the following geometric sequences of numbers. Could you have figured out what the common ratio is in each sequence?

 Sequence

 Common Ratio
(Fixed Number Being Multiplied)

3,12,48,192..

 4

32,-16,8,-4, ...

-0.5 

 1,3,9,27,81,...

 3

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.

 

Check for Understanding

Answer the following questions about geometric sequences.

Enter your answer and press tab after each response

Graded Response

 a) Find the common ratio in the sequence
28, 14, 7, 3.5, ...
a)
 b) The first term in a geometric sequence is -1 and the common ratio is 2. What is the fifth term in the sequence ?
b)

External Links to Extend Basic Understanding

Harcourt's Interactive Math Glossary: Read another defintion of geometric sequence.

Related Investigations on InterMath to Challenge Teachers

Geometric Sequences and Series: Explore the properties of geometric sequences and series.
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.

Top of Page

 


Fibonacci Sequence

Fibonacci Sequence - The sequence 1, 1, 2, 3, 5, 8, 13, 21, ..., where each term is the sum of the two preceeding terms.

 

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 Numbers??
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 Sequence.

 

Check for Understanding

Answer the following questions about the Fibonacci sequence:

Enter your answer and press tab after each response

Graded Response

 a) What number follows 21 in this sequence ?
a)
 b) Which of the following is not a real world example of the numbers in this sequence ?
b)

 

External Links to Extend Basic Understanding

Harcourt's Interactive Math Glossary: Read another definition of the Fibonacci sequence.

Related Investigations on InterMath to Challenge Teachers

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 recursively.
Ratio of Fibonacci
: Use the Fibonacci sequence to examine ratios.

Top of Page


How-To Page | Welcome Page | Main InterMath Page