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Sequences and Series (page 1 of 5)

Sections: Terminology and notation, Basic examples, Arithmetic and geometric sequences, Arithmetic series, Finite and infinite geometric series

A "sequence" (or "progression", in British English) is an ordered list of numbers; the numbers in this ordered list are called "elements" or "terms". A "series" is the value you get when you add up all the terms of a sequence; this value is called the "sum". For instance, "1, 2, 3, 4" is a sequence, with terms "1", "2", "3", and "4"; the corresponding series is the sum "1 + 2 + 3 + 4", and the value of the series is 10.

A sequence may be named or referred to as "A" or "An". The terms of a sequence are usually named something like "ai" or "an", with the subscripted letter "i" or "n" being the "index" or counter. So the second term of a sequnce might be named "a2" (pronounced "ay-sub-two"), and "a12" would designate the twelfth term.

Note: Sometimes sequences start with an index of n = 0, so the first term is actually a0. Then the second term would be a1. The first listed term in such a case would be called the "zero-eth" term. This method of numbering the terms is used, for example, in Javascript arrays. Don't assume that every sequence and series will start with an index of n = 1.

A sequence A with terms an may also be referred to as "{an}", but contrary to what you may have learned in other contexts, this "set" is actually an ordered list, not an unordered collection of elements. (Your book may use some notation other than what I'm showing here. Unfortunately, notation doesn't yet seem to have been entirely standardized for this topic. Just try always to make sure, whatever resource you're using, that you are clear on the definitions of that resource's terms and symbols.)




To indicate a series, we use either the Latin capital letter "S" or else the Greek letter corresponding to the capital "S", which is called "sigma" (SIGG-muh):


To show the summation of, say, the first through tenth terms of a sequence {an}, we would write the following:

    sigma; "n = 1" below, "10" above, and "a-sub-n" to the right

The "n = 1" is the "lower index", telling us that "n" is the counter and that the counter starts at "1"; the "10" is the "upper index", telling us that a10 will be the last term added in this series; "an" stands for the terms that we'll be adding. The whole thing is pronounced as "the sum, from n equals one to ten, of a-sub-n". The summation symbol above means the following:

    a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10

The written-out form above is called the "expanded" form of the series, in contrast with the more compact "sigma" notation. Copyright Elizabeth Stapel 2006-2011 All Rights Reserved

Any letter can be used for the index, but i, j, k, and n are probably used more than any other letters.

Sequences and series are most useful when there is a formula for their terms. For instance, if the formula for an is "2n + 3", then you can find the value of any term by plugging the value of n into the formula. For instance, a8 = 2(8) + 3 = 16 + 3 = 19. In words, "an = 2n + 3" can be read as "the n-th term is given by two-enn plus three". The word "n-th" is pronounced "ENN-eth", and just means "the generic term an, where I haven't yet specified the value of n."

Of course, there doesn't have to be a formula for the n-th term of a sequence. The values of the terms can be utterly random, having no relationship between n and the value of an. But sequences with random terms are hard to work with and are less useful in general, so you're not likely to see many of them in your classes.

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Cite this article as:

Stapel, Elizabeth. "Sequences and Series." Purplemath. Available from Accessed


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