Sequences and Series- An introduction

Sequence:


Let us consider the following collection of numbers – 
(1) 28,2,25,27,--------
(2) 2,7,11,19,31,51, ---------
(3) 1,2,3,4,5,6, --------------
(4) 20.5,18.5,16.5,14.5,12.5,10.5, ------------
In (1) the numbers are not arranged in a particular order. In (2) the numbers are in ascending order but they do not obey any rule or law. It is, therefore not possible to indicate the number next to 51.
In (3) we find that by adding 1 to any number, we get the next number. So the number after 6 would be = 6+1 = 7.
In (4) if we subtract 2 from any number we get the number that follows. So here the number after 10.5 would be = 10.5-2 = 8.5
Under these circumstances, we say, the numbers in the collections (1) and (2) do not form sequences whereas the numbers in the collections (3) and (4) form sequences.
Definition: An ordered collection of numbers a1,a2,a3,a4,….. an,…… is called a sequence if according to some definite rule or law there is a definite value of an, called the term or element of the sequence, corresponding to any value of the natural number n.
Clearly the nth term of a sequence is a function of the positive integer n. If the nth term itself is also always an integer, then such a sequence is called an integer sequence.

Series:

An expression of the form a1+a2+a3+a4+….. +an+….. which is the sum of the elements of a sequence {an} is called a series.

If Sn = u1+u2+u3+….un, the Sn is called the sum to n terms (or the sum of first n terms) of the series and is denoted by the Greek letter sign ?.
Thus Sn = ?_(r=1)^n¦u_r

If a sequence or a series contains finite number of elements, it is called a finite sequence or series, otherwise they are called infinite sequences and series.

Progressions:

There are mainly three types of progressions –
(a) Arithmetic progression (A.P.): That means a sequence in which each term is obtained by adding a constant d to the preceding term. This constant ‘d’ is called the common difference of the arithmetic progression.
(b) Geometric progression (G.P.): If in a sequence of terms each term is constant multiple of the proceeding term, then the sequence is called a Geometric Progression (G.P.). The constant  multiplier is called the common ratio (r).
(c) Harmonic progression (H.P.): If each term of an A.P. is replaced by its reciprocal, then we get a harmonic progression.