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When a collection of objects is listed in a sequential manner or a definite order such that every member has a definite position, that is, it comes either before or after, every other member, it is called a sequence.

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

The members or numbers that are listed in the sequence are called its terms.

The terms of a sequence are denoted by a_{1}, a_{2}, a_{3},â€¦,a_{n},â€¦

The n^{th} term of a sequence is denoted by a_{n}. a_{n} is also known as the general term of the sequence.

**Finite or infinite sequence:**

If the number of terms in a sequence is finite or countable, then it is called a finite sequence.

If the sequence goes on forever or has an uncountable number of terms, then it is called an infinite sequence.

Ex: 1, 2, 3, 4, ... is an infinite sequence. 2, 4, 6, 8, 10 is a finite sequence.

Terms of a sequence can be expressed by an algebraic formula.

Consider the sequence 1/2, 1/4, 1/8, 1/16,....

The first term can be written as **a _{1} = 1/2 = 1/2^{1}**;

The second term can be written as **a _{2} = 1/4 = 1/2^{2}**;

The third term can be written as **a _{3} = 1/8 = 1/2^{3}** ;

The fourth term can be written as **a _{4} = 1/16 = 1/2^{4}**, and so on.

n^{th} term of this sequence can be expressed as **a _{n} = 1/2^{n}**, where n is a natural number.

Any term of the sequence can be found by substituting the term value in place of n.

e.g. **a _{8} = 1/2^{8}**

In some cases, an arrangement of numbers in a sequence like {1,1, 3, 7, 17, 41, â€¦â€¦â€¦} has no apparently visible pattern, but the terms of this sequence are generated by the relation given by

a_{1} = a_{2} =1

a_{3} = a_{1} + 2a_{2};

a_{4} = a_{2} + 2a_{3}, and so on.

The nth term of this sequence is **a _{n} = a_{n-2} + 2a_{n-1}**, n>2.

Sequence {1, 1, 2, 3, 5, 8 â€¦} can be described as:

a_{1} = a_{2} =1;

a_{3} = a_{1} + a_{2};

a_{4} = a_{2} + a_{3} and **a _{n} = a_{n-2} + a_{n-1}**, n >2.

This sequence is known as the Fibonacci sequence.

Sometimes, it is practically impossible to express the terms of a sequence by means of a particular algebraic formula. Such sequences can only be described verbally, like the sequence of prime numbers {2,3,5,7,â€¦â€¦â€¦..}.

A sequence is a collection of all images of a function, x, from **N** or any subset of **N** to **X**, where **N** is the set of all natural numbers and **X** is any set, may be of objects, events or numbers.

Domain of function x = **N**

{ x(1), x(2), x(3), x(4), â€¦} Or **{x _{1},x_{2},x_{3},x_{4},...}**, where x(1)= x

Sequences are also called progressions because the terms in a sequence progress in a definite manner obeying some specific algebraic rule.

**Series**

Let {a_{1}, a_{2}, a_{3}, a_{4,...}} be a sequence.

Then the expression **a _{1} + a_{2} + a_{3 }+ a_{4} + ... + a_{n}**

The expression obtained on adding the terms of a sequence is called a series.

Here, **a _{1 }**is called the first term,

A series is finite if the given sequence is finite, and infinite if the given sequence is infinite.

In summation notation, the infinite series **a _{1} + a_{2} + a_{3 }+ a_{4} + ... + a_{n}**

Similarly, in summation notation, the finite series **a _{1} + a_{2} + a_{3 }+ a_{4} + ... + a_{n}**

âˆ‘^{âˆž}_{n=1}a_{n }or âˆ‘^{âˆž}_{k=1}a_{k} are known as the compact forms of a series.

When a collection of objects is listed in a sequential manner or a definite order such that every member has a definite position, that is, it comes either before or after, every other member, it is called a sequence.

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

The members or numbers that are listed in the sequence are called its terms.

The terms of a sequence are denoted by a_{1}, a_{2}, a_{3},â€¦,a_{n},â€¦

The n^{th} term of a sequence is denoted by a_{n}. a_{n} is also known as the general term of the sequence.

**Finite or infinite sequence:**

If the number of terms in a sequence is finite or countable, then it is called a finite sequence.

If the sequence goes on forever or has an uncountable number of terms, then it is called an infinite sequence.

Ex: 1, 2, 3, 4, ... is an infinite sequence. 2, 4, 6, 8, 10 is a finite sequence.

Terms of a sequence can be expressed by an algebraic formula.

Consider the sequence 1/2, 1/4, 1/8, 1/16,....

The first term can be written as **a _{1} = 1/2 = 1/2^{1}**;

The second term can be written as **a _{2} = 1/4 = 1/2^{2}**;

The third term can be written as **a _{3} = 1/8 = 1/2^{3}** ;

The fourth term can be written as **a _{4} = 1/16 = 1/2^{4}**, and so on.

n^{th} term of this sequence can be expressed as **a _{n} = 1/2^{n}**, where n is a natural number.

Any term of the sequence can be found by substituting the term value in place of n.

e.g. **a _{8} = 1/2^{8}**

In some cases, an arrangement of numbers in a sequence like {1,1, 3, 7, 17, 41, â€¦â€¦â€¦} has no apparently visible pattern, but the terms of this sequence are generated by the relation given by

a_{1} = a_{2} =1

a_{3} = a_{1} + 2a_{2};

a_{4} = a_{2} + 2a_{3}, and so on.

The nth term of this sequence is **a _{n} = a_{n-2} + 2a_{n-1}**, n>2.

Sequence {1, 1, 2, 3, 5, 8 â€¦} can be described as:

a_{1} = a_{2} =1;

a_{3} = a_{1} + a_{2};

a_{4} = a_{2} + a_{3} and **a _{n} = a_{n-2} + a_{n-1}**, n >2.

This sequence is known as the Fibonacci sequence.

Sometimes, it is practically impossible to express the terms of a sequence by means of a particular algebraic formula. Such sequences can only be described verbally, like the sequence of prime numbers {2,3,5,7,â€¦â€¦â€¦..}.

A sequence is a collection of all images of a function, x, from **N** or any subset of **N** to **X**, where **N** is the set of all natural numbers and **X** is any set, may be of objects, events or numbers.

Domain of function x = **N**

{ x(1), x(2), x(3), x(4), â€¦} Or **{x _{1},x_{2},x_{3},x_{4},...}**, where x(1)= x

Sequences are also called progressions because the terms in a sequence progress in a definite manner obeying some specific algebraic rule.

**Series**

Let {a_{1}, a_{2}, a_{3}, a_{4,...}} be a sequence.

Then the expression **a _{1} + a_{2} + a_{3 }+ a_{4} + ... + a_{n}**

The expression obtained on adding the terms of a sequence is called a series.

Here, **a _{1 }**is called the first term,

A series is finite if the given sequence is finite, and infinite if the given sequence is infinite.

In summation notation, the infinite series **a _{1} + a_{2} + a_{3 }+ a_{4} + ... + a_{n}**

Similarly, in summation notation, the finite series **a _{1} + a_{2} + a_{3 }+ a_{4} + ... + a_{n}**

âˆ‘^{âˆž}_{n=1}a_{n }or âˆ‘^{âˆž}_{k=1}a_{k} are known as the compact forms of a series.