In mathematics, the word, "sequence" is used in much the same way as it is in ordinary English. When we say that a collection of objects is listed in a sequence, we usually mean that the collection is ordered in such a way that it has an identified first member, second member, third member and so on.
For example, population of human beings or bacteria at different times form a sequence. The amount of money deposited in a bank, over a number of years form a sequence. Depreciated values of certain commodity occur in a sequence. Sequences have important applications in several spheres of human activities.
Sequences, following specific patterns are called progressions.
By a sequence, we mean an arrangement of number in definite order according to some rule. Also, we define a sequence as a function whose domain is the set of natural numbers or some subsets of the type {1, 2, 3, ...., k}. A sequence containing a finite number of terms is called a finite sequence. A sequence is called infinite if it is not a finite sequence.
Let a1, a2, a3, ... be the sequence, then the sum expressed as a1 + a2 + a3 + ... is called series. A series is called finite series if it has got finite number of terms.
An arithmetic progression (A.P.) is a sequence in which terms increase or decrease regularly by the same constant. This constant is called common difference of the A.P.
Usually, we denote the first term of A.P. by a, the common difference by d and the last term by l. The general term or the nth term of the A.P. is given by an = a + (n - 1)d.
The arithmetic mean A of any two numbers a and b is given by (a + b)/2 i.e., the sequence a, A, b is in A.P.
A sequence is said to be a geometric progression or G.P., if the ratio of any term to its preceding term is same throughout. This constant factor is called the common ratio. Usually, we denote the first term of a G.P. by a and its common ratio by r. The general or the nth term of G.P. is given by an = arn-1.
The geometric mean (G.M.) of any two positive numbers a and b is given by √(ab) i.e., the sequence a, G, b is G.P.