Sequence

Sequence:

Let us understand the meaning of Sequence in general,in mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and the exact same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function.Let us now look at a more diversified explanation about a Sequence,A set of numbers arranged in a definite order according to some definite rule is called a sequence.
or

A sequence is a function whose domain is the set N of natural numbers.
It is customary to denote a sequence by a letter 'a' and the image a(n) or t(n), n Î N under 'a' by an or tn.
Examples:

1, 3, 5, 7…..... (adding 2 to every term)

1, 4, 16, 64 … (Multiplying by 4 every term)

20, 17, 14 … . (add -3 to every term)

The different numbers in a sequence are called terms of sequence.

The subscripts denote the position of the term.

In the second example, 4 is the second term, and 14 is the third term in the third example.

The nth term of a sequence is called the general term of the sequence and is usually denoted by an or tn.

Finite and Infinite Sequences:

A sequence is called finite if the number of terms is finite. A finite sequence has always a last term.
Examples:

2, 5, 8, 11, 14 …, 32
37, 33 …, 1

A sequence is called infinite if the number of terms is infinite. An infinite sequence has no last term. In this sequence, every term is followed by a new term.
Examples:

i) A sequence of multiples of 5
5, 10, 15, 20, …

ii) A sequence of reciprocals of positive integers

The above two sequences are clearly the infinite sequences.
Series

Indicated sum of the terms in a sequence is called a series. The result of performing the additions is the sum of the series.

Examples:

i) 1 + 4 + 7 + 10 + ... is a series in which first term is 1, second term is 4, third term is 7 and so on.
ii) 3 - 9 + 27 - 81 + ... is also a series in which the first term is 3, second term is -9, third term is 27 and so on.


Hope you like the above example of Sequence.Please leave your comments, if you have any doubts.