The arithmetic sequence is a pattern of numbers in which the difference between any two consecutive terms stays the same. A sequence is a collection of numbers that follows a rule or a pattern. For example, the sequence 1, 6, 11, 16,.. is an arithmetic sequence because there is a pattern where each number is obtained by adding 5 to the previous term, and this pattern is assumed to continue forever.
There are all types of arithmetic sequences. Some of them are shown below.
The common difference of an arithmetic sequence is the constant difference between consecutive terms. A common difference can be a negative or positive number.
Example 1
What is the common difference in the following sequence?
The common difference is 11, as each consecutive term can be reached by adding 11 to the previous term.
Example 2
What is the common difference in the following sequence?
The common difference is -4 because each consecutive term can be reached by adding -4 to the previous term.
When working with arithmetic sequences, we use the term "n" to indicate the term number and " " to indicate the term.
So, if you need to find in the sequence the answer is 7, because 7 is the fourth term in the sequence.
Similarly, for any term number , ( ) represents the term that comes before the term.
Recursive formulas of arithmetic sequences provide two pieces of information:
For example,
This tells us that the first term is 3
This tells us that the term is found by adding 2 to the previous term.
Example 3
So to find the 5th term, for example, we need to extend the sequence term by term:
The explicit formula allows you to simply plug in the number of the term you are interested in to get the value of that term, rather than having to write out each step as with recursive formulas. This is the recursive formula for the arithmetic sequence 3, 5, 7 …
Example 4
So to find the term, we need to plug into the explicit formula.
The formulas used for arithmetic sequences work like functions-you input term number , and the formula outputs the value of that term .
Sequences are actually defined as functions. But you should note that n cannot be any real number value. There's no such thing as a negative fifth term, for example, or the term of a sequence.
In fact, the domain of sequences, which is the set of all possible inputs of the function, is the positive integers.
a. Find in the sequence where and
The first term is -5
Add 9 to the previous term
So
b. Find in the sequence given by the function
Sum of the First n Terms of an Arithmetic Sequence
College Algebra Diagnostic Tests
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