Sequences

A sequence of terms from a set S is a function from some set {1, 2, . . . } of positive integers into the set S.

Informally, a sequence is a list of terms, usually numbers. The terms in the sequence may repeat.

A sequence is finite if the function's domain is a finite set {1, 2, . . . n}. A finite sequence of n terms is also known as an ordered n-tuple. A sequence that's not finite is an infinite sequence.

Incidentally, the above definition of sequence illustrates how a mathematical function can be thought of as "picking out" elements of some set. In this case, the function is putting those chosen elements into an order.

One common kind of sequence is the arithmetic sequence, whose terms are numbers, and in which the difference between consecutive terms is constant. That is, you get the same result if you subtract the second term from the third, or the third from the fourth, or the forty-ninth from the fiftieth. Examples of arithmetic sequences include

{1, 5, 9, 13, 17, 21, 25, . . . }

{0.5, 1.5, 2.5, 3.5, 4.5}

{Ð3, Ð3, Ð3, Ð3, Ð3, . . . .}

{2¹, ¹, 0, Ð¹, Ð2¹}.

Another common kind of sequence is a geometric sequence, also of numbers, in which the quotient of consecutive numbers is constant. Some geometric sequences include
{1, 2, 4, 8, 16, . . . }

{Ð3, Ð1, Ð1/3, Ð1/9, Ð1/27}

{2, 2, 2, 2, 2, 2, . . . }.

Sequences of letters from the set {a, b, c, d, e} include
{a, a, b, c}

{e}

{d, d, d, d, d, d, . . .}.