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**Basic Definitions of Sequences and Series**

In Number sense, A set of numbers arranged in a definite order according to some definite rule is called a **sequence**. Sequences have wide applications. **For example**, the amount of money in a fixed deposit in a bank, over a number of years increases in sequence.

A sequence is a function whose domain is the set N of natural numbers.

Or

A set of numbers arranged in a definite order according to some definite rule is called a sequence.

It is customary to denote a sequence by a letter 'a' and the image a(n) or t(n), n $in$ N, under ‘a’ by a_{n} or t_{n}.

Examples:

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

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

20, 17, 14 …. (Add -3 to every term)

The different numbers in a geometric sequence are called **terms** of the sequence. These terms are usually denoted by a_{1}, a_{2}, a_{3}…a_{n} or t_{1}, t_{2}, t_{3}…t_{n}.

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 n^{th }term of a sequence is called the general term of the sequence and is usually denoted by a_{n} or t_{n}.

In Sequences and Series , A arithmetic sequence is called **finite** if the number of terms is finite. A finite sequence always has 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.

Example 1: A sequence of multiples of 5

5, 10, 15, 20…..

Example 2: A sequence of reciprocals of positive integers

1, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, ........

The above two sequences are clearly infinite sequences. We cover more Problems on Sequence and series, which helps us to understand the concept better.

In Sequences and Series , A Sequence is said to be a progression if the absolute value of the terms increases (or decreases).

In Sequences and Series, Indicated sum of the terms in a sequence is called a geometric series . The result of performing the additions is the sum of the series.

The indicated sum a_{1}+a_{2}+a_{3}+…+a_{n} of the terms of a sequence a_{1}, a_{2}, a_{3},…a_{n} is called a geometric series.

Example : 1 + 4 + 7 + 10 + ... is a series

Here the first term is 1, second term is 4, third term is 7 and so on.

Integer Sequences are a kind of sequences.It is explicity given by its `n` ^{th} term and implicity given by the relationship of its terms.

For any given integers a[0] and a [1] , an infinite sequence can be given as:

a [ k + 2 ] = ( 2k + 3) a[k + 1] + (k + 1)^{2} a[k]

There are different examples of Integer sequences like the Fibonacci sequences, Kolakoski sequences etc,.