Mathematics Progression & Series Introduction

Sequence

A succession of numbers arranged in a definite order or arrangement according to some well-defined law is called a sequence.
OR
A sequence is a function of natural numbers (N) with co-domain is the set of real numbers (R) (complex numbers (C)]. If range is subset of real numbers (complex numbers), then it is called a real sequence (complex sequence).
OR
A mapping `f: N -> C,` then `f(n) = t_n n in N` is called a sequence to he denoted it by `{f(1),f(2), f(3), ... } = { t_1, t_2 , t_3, ... } == { t_n }.`
The `nth` term of a sequence is denoted by `T_n, t_n ,a_n, a(n), u_n` etc.

Remark The sequence `a_1, a_2 ,a_3 ......` is generally written as `{ a_n }.`

i.e. 1, 2, 3, 5, 8, 13, ... is a sequence, because each term (except first two) is obtained by taking the sum of preceding two terms.

`text(Types of Sequences)`

There are two types of sequences.

`1.` `text(Finite Sequence)`
A sequence is said to be finite sequence, if it has finite number of terms. A
finite sequence is described by `a_1 ,a_2 , a_3 , ... a_n` or `T_1 , T_2 , T_3 ,...........T_n` where `n in N.`

`text(Illustrations)` (i) 3, 5, 7, 9, ... , 37 (ii) 2, 6, 18, 54, ... , 4374

`2.` `text(Infinite Sequence)`
A sequence is said to be an infinite sequence, if it has infinite number of
terms. An infinite sequence is described by `a_1 ,a_2 , a_3 , ... ` or `T_1 , T_2 , T_3 ,...........`
`text(Illustrations)` `(i) 1. 1/3 , 1/9 , 1/29 , ............ (ii) 1, 1/2 , 1/4 , 1/8 , 1/16 , ..............`

Series

In a sequence, the sum of the directed terms is called a series.
For example, If 1, 4, 7, 10, 13, 16, ... is a sequence, then its sum i.e.,
1 + 4 + 7 + 10 + 13 + 16 + ... is a series.

In general, if `T_1, T_2 , T_3 , ... , T_n , ...` denote a sequence, then the symbolic
expression `T_1 + T_2 + T_3 + ... + T_n + ...` is called a series associated with the given
sequence.

Each member of the series is called its term.
In a series `T_1 + T_2 + T_3 + ... + T_r + ... ,` the sum of first n terms is denoted by `S_n.`
Thus,
`S_n = T_1 + T_2 + T_3 + ... + T_n = sum_(r=1)^n T_r = sum T_n `
If `S_n` denotes the sum of n terms of a sequence.
Then, `S_n - S_(n -1_ = (T_1 + T_2 + T_3 + ... + T_n)- (T_1 + T_2 + ... + T_(n -1) = T_n`
Thus, `T_n = S_n - S_(n - 1)`

`text(Types of Series)`

There are two types of series.

`1.` `text(Finite Series)`

A series having finite number of terms is called a finite series.
`text(Illustrations)`
(i) 3 + 5 + 7 + 9 + ... + 21
(ii) 2 + 6 + 18 +54+ ... + 4374

`2.` `text(Infinite Series)`
A series having an infinite number of terms is called an infinite series.
`text(Illustrations)`
`(i) 1+1/3 + 1/9 + 1/27 + ........`
`(ii) 1+ 1/2 + 1/4 + 1/8 + ...........`

Progression

If the terms of a sequence can be described by an explicit formula, then the sequence is called a progression.
OR
A sequence is said to be progression, if its terms increases (respectively decreases) numerically.

For example, The following sequences are progression :
`(i) 1, 3, 5, 7,...` `(ii) 1/2 , 1/6 , 1/18 , 1/54 , ............` `(iii) 1 , -1/3 , 1/9 , -1/27 , ..........`
`(iv) 1, 8 , 27 , 256 , ............` `(v) 8 , -4 , -1 , 1/2, ...........`

The sequences (iii) and (v) are progressi

`|1| > |-1/3| > |1/9| > |-1/27| > ..........`

i.e., `1 > 1/3 > 1/9 > 1/27 ............`

and `|8| > |-4| < |2| > |-1| > |1/2| >..........`

i.e., `8 > 4 > 2 > 1 > 1/2 > ..................`

`text(Note)`
all the definition and formulae are valid for complex numbers in the thoery of progression but should be assumed (if not otherwise stated) that the terms of the progressions are real numbers.