A sequence is a list of numbers. Any time you write numbers in a list format, you are creating a sequence. Something as simple as
1, 2, 3, 4, 5, 6, . . .
is a sequence.
Rather than just listing the numbers, we usually identify it as a sequence
with the notation
an = 1, 2, 3, 4, 5, 6, . . .
Usually there is some type of pattern
to a sequence. In the sequence
above, you are adding one to each term to get the next term.
of a sequence
is just a number that is in the sequence.
Terms can be identified by their location. We note the 1st
term in a sequence
and we would call the 5th
term in the sequence
We described the pattern
in the sequence
as adding one to each term to get the next term. We can express this as a recursive formula by writing
an = an-1 + 1
This says to get any term in the sequence
), add one (+1) to the previous term (an-1
A recursive formula is written in such a way that in order to find any term in a sequence, you must know the previous terms. In other words, to find the 12th term, you would need to know the first 11. There are times when this can be a difficult task and there will be other ways to write sequences. But it is important to know that many sequences are best described using recursive formulas.
The simple sequence
we have been looking at is called an arithmetic sequence
. Any time you are adding the same number to each term to complete the sequence, it is called an arithmetic sequence. The number that is added to each term is called the common difference
and denoted with the letter d
. So in our example we would say that d = 1. The common difference
can be subtracting two consecutive terms. You can subtract any two terms as long as they are consecutive. So we could find d by taking 5 - 4 = 1 or 2 - 1 = 1. Notice that we will always use the term that appears later in the sequence
first and then subtract the term that is right in front of it.
If we looked at a sequence
= 1, 3, 9, 27, 81, 243, . . . this would not fit our definition of an arithmetic sequence. We are not adding the same number to each term. However, notice that we are multiplying each term by the same number (3) each time. When you multiply every term by the same number to get the next term in the sequence, you have a geometric sequence
. Geometric sequences can also be written in recursive form. In this case, we would write
. Remember that in the language of sequences we are saying, to find any term in the sequence
), multiply the previous term (bn-1
) by 3.
Just as arithmetic sequences have a common difference, geometric sequences have a common ratio
which is denoted with the letter r
. The common ratio
is found by dividing successive terms in the sequence. So in our geometric sequence
example, we could use 9/3 = 3 or 243/81=3 to find that r = 3. As with finding a common difference, when we find a common ratio, we must use the term that appears later in the sequence
as our numerator and the number right before it as our denominator.
There are other types of sequences that do not fit into the arithmetic or geometric category, but are still considered sequences because there is a pattern
to determining the next term. Our focus in these lessons will be on arithmetic and geometric. For more information on other types of sequences, ask your teacher.
As mentioned earlier, recursive forms of sequences have their drawbacks, but are a useful way to see what is happening in a sequence. The other way to write sequences is called an explicit form or a closed form. These will be explored in other lessons.
Another topic associated with sequences is series. A series
is simply adding the terms in a sequence. An arithmetic series
involves adding the terms of an arithmetic sequence
and a geometric series
involves adding the terms of a geometric sequence. These will be explored in other lessons.
This lesson has provided an introduction to the terminology needed to continue working with sequences and series. One important skill is being able to identify what type of sequence
you have. Do the “Try These” below and after successful completion of these problems, continue with other lessons on sequences and series.