Introduction to Sequences
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Examples, solutions, videos, worksheets, and activities to help Algebra II students learn about sequences.
The following diagram defines and give examples of sequences: Arithmetic Sequences, Geometric Sequences, Fibonacci Sequence. Scroll down the page for more examples and solutions using sequences.
Arithmetic Sequence
An arithmetic sequence (also known as an arithmetic progression) is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by ’d'.
The n-th Term of an Arithmetic Sequence
This formula allows you to find any term in the sequence without having to list all the preceding terms.
The formula for the n-th term (an) is:
an = a1 + (n − 1)d
Where:
an = the n-th term of the sequence (the term you want to find)
a1 = the first term of the sequence
n = the position of the term in the sequence
d = the common difference between consecutive terms
Geometric Sequence
A geometric sequence (also known as a geometric progression) is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, denoted by ‘r’.
The n-th Term of a Geometric Sequence
This formula allows you to find any term in the sequence without having to list all the preceding terms.
The formula for the n-th term (an) is:
an = a1rn−1
Where:
an = the n-th term of the sequence (the term you want to find)
a1 = the first term of the sequence
n = the position of the term in the sequence
r = the common ratio
Fibonacci Sequence
The Fibonacci Sequence is a famous sequence of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1.
Definition and Rule:
The sequence typically begins:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
The rule to generate the sequence is:
Fn = Fn-1 + Fn-2
Where:
Fn is the n-th term in the sequence.
Fn-1 is the term immediately preceding Fn.
Fn-2 is the term two places preceding Fn.
Videos
Introduction to Sequences
Lists of numbers, both finite and infinite, that follow certain rules are called sequences. This introduction to sequences covers the definition of a sequence and how to identify a rule. There are specific sequences that have their own formulas and methods for finding the value of terms, such as arithmetic and geometric sequences. Series are an important concept that come from sequences.
How to define a sequence?
Define a sequence.
Determine whether a sequence is arithmetic or geometric
What is a Sequence?
Basic Sequence Information. This video discusses what a sequence is, what it means for a sequence to converge or diverge, and do some examples.
Introduces the concept of determining if sequence converges or diverges.
Number Sequences
The relationship between the sequence of squares and the sequence of odd numbers.
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