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Patterns are also very common in the process of making things. Consider a car factory that bases its production of cars following a specific pattern. In their case, they have planned for the number of cars they produce to increase by the same amount each month.
Patterns like the given scenarios can be modeled using sequences. This lesson demonstrates these sequences and explores a special type called an arithmetic sequence.
Here are a few recommended readings before getting started with this lesson.
The following applet shows the firsts five terms of an infinite sequence. Analyze them carefully and determine whether or not the sequence is arithmetic.
Every arithmetic sequence can be described by a linear function that is defined for the set of counting numbers. This function, referred to as the explicit rule of an arithmetic sequence, follows a specific general format.
an=a1+(n−1)d
n | an | Rewrite |
---|---|---|
1 | a1 | a1+0⋅d |
2 | a2 | a1+1⋅d |
3 | a3 | a1+2⋅d |
4 | a4 | a1+3⋅d |
5 | a5 | a1+4⋅d |
The coefficient of the common difference is always 1 less than the value of the position n. This makes it possible to write an explicit rule like the following formula.
an=a1+(n−1)d
Davontay goes to a cinema to see a new movie about a kid who runs like the wind. Waiting for the movie to start, he notices something very interesting about the theater itself.
n=50
Subtract term
Multiply
Add terms
Davontay, inspired by a movie about running, downloaded an app that keeps track of his total running distance. It even predicts his running goals. He went for a long run on his first day using it and has followed that same route ever since. One day, he opened the app, but the free trial period was over, and his data was locked!
n=23
a23=60
Subtract term
Multiply
LHS−55=RHS−55
Rearrange equation
an=150
LHS−5=RHS−5
LHS/2.5=RHS/2.5
LHS+1=RHS+1
Rearrange equation
The following applet shows two of the first five terms of an infinite arithmetic sequence. Analyze the sequence carefully and determine which of the three options is the correct explicit rule of the sequence.
The following applet shows the first five terms of an infinite arithmetic sequence. Determine the explicit rule of the sequence and use it to find the indicated term.
It has been shown how an explicit rule describes an arithmetic sequence. This representation uses a formula that receives the term position as the input and returns the term's value as the output. However, an arithmetic sequence can also be described by using a recursive rule.
A recursive rule of an arithmetic sequence is a pair comprised of a recursive equation telling how the term an is related to its preceding term an−1 and the initial term of the sequence a1.
a1,an=an−1+d
Now it will be explained, step by step, how to write the recursive rule for an arithmetic sequence and how to use it to find an unknown term's value.
The recursive rule of an arithmetic sequence gives the first term of the sequence and a recursive equation.
a1,an=an−1+d