Arithmetic Sequences and Series
This page will teach you about arithmetic sequences and series. Here are the sections within this page:
Sequences of numbers that follow a pattern of adding a fixed number from one term to the next are called arithmetic sequences. The following sequences are arithmetic sequences:
Sequence A:   5 , 8 , 11 , 14 , 17 , ...
For sequence A, if we add 3 to the first number we will get the second number. This works for any pair of consecutive numbers. The second number plus 3 is the third number: 8 + 3 = 11, and so on.
For sequence B, if we add 5 to the first number we will get the second number. This also works for any pair of consecutive numbers. The third number plus 5 is the fourth number: 36 + 5 = 41, which will work throughout the entire sequence.
Sequence C is a little different because we need to add -2 to the first number to get the second number. This too works for any pair of consecutive numbers. The fourth number plus -2 is the fifth number: 14 + (-2) = 12.
Because these sequences behave according to this simple rule of addiing a constant number to one term to get to another, they are called arithmetic sequences. So that we can examine these sequences to greater depth, we must know that the fixed numbers that bind each sequence together are called the common differences. Sometimes mathematicians use the letter d when referring to these types of sequences.
Mathematicians also refer to generic sequences using the letter a along with subscripts that correspond to the term numbers as follows:
This means that if we refer to the fifth term of a certain sequence, we will label it a5. a17 is the 17th term. This notation is necessary for calculating nth terms, or an, of sequences.
The d-value can be calculated by subtracting any two consecutive terms in an arithmetic sequence.
...where n is any positive integer greater than 1.
In order for us to know how to obtain terms that are far down these lists of numbers, we need to develop a formula that can be used to calculate these terms. If we were to try and find the 20th term, or worse to 2000th term, it would take a long time if we were to simply add a number -- one at a time -- to find our terms.
If a 5-year-old was asked what the 301st number is in the set of counting numbers, we would have to wait for the answer while the 5-year-old counted it out using unnecessary detail. We already know the number is 301 because the set is extremely simple; so, predicting terms is easy. Upon examining arithmetic sequences in greater detail, we will find a formula for each sequence to find terms.
It may be necessary to calculate the number of terms in a certain arithmetic sequence. To do so, we would need to know two things.
We would need to know a few terms so that we could calculate the common difference and ultimately the formula for the general term. We would also need to know the last number in the sequence.
Once we know the formula for the general term in a sequence and the last term, the procedure is relatively uncomplicated. Set them equal to each other. Since the formula uses the variable n to calculate terms, we can also use it to determine the term number for any given term.
ideo: Finding the Number of Terms in a Finite Arithmetic Sequence
Given our generic arithmetic sequence...
...we can add the terms, called a series, as follows.
Given the formula for the general term an = dn + c, there exists a formula that can add such a finite list of these numbers. It requires three pieces of information. The formula is...
...where Sn is the sum of the first n numbers, a1 is the first number in the sequence and an is the nth number in the sequence.
If you would like to see a derivation of this arithmetic series sum formula, watch this video.
Usually problems present themselves in either of two ways. Either the first number and the last number of the sequence are known or the first number in the sequence and the number of terms are known.
Watch these instructional videos.
ideo: Finding the nth Term of an Arithmetic Sequence
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