sequences and series

A sequence is a set of terms which are written in a definite order obeying certain rules e.g. 2,4,6,8,10,... is an infinite (since it goes on and on forever) sequence obeying a certain rule which is; for you to get the next term in the sequence you add 2 to the preceding one.In particular this sequence is made of multiples of two or even better the even numbers.

A sequence with definite terms is said to be a finite sequence e.g. 3,6,9,12. Note that they are no three dots after 12 i.e. the three dots normally denote that the sequence goes on and on forever. Hence this sequence has 4 terms. Hence  it is a finite sequence.

A series is the sum of the terms of a sequence e.g. 2+4+6+8+10+... is a series. In general a finite series is the sum of terms of a finite sequence i.e. a1+a2+a3+...+an. While an infinite series is the sum of terms of an infinite sequence i.e. a1+a2+a3+...+an+....

The most common sequences are the arithmetic sequences and the geometric sequences. An arithmetic sequence is a sequence that proceeds with a common difference which is normally denoted by d.In general an arithmetic sequence is a sequence of the form; a, [a+d], [a+2d],[a+3d],...,[a+(n-1)d] where a is the first term, n  is the number of terms in the sequence and [a+(n-1)d] is the last term which is either denoted as l or Tn (i.e. the nth term). For example; 2,4,6,8,10,... is an arithmetic sequence with first term a=2 and common difference d=2.

An Arithmetic series also referred to as an Arithmetic progression is the sum of terms of an arithmetic sequence i.e. a+[a+d]+[a+2d]+[a+3d]+...+[a+(n-1)d] e.g. 4+7+10+13+...is an infinite arithmetic series with first term a=4 and common difference d=3


On the other hand a geometric sequence is a sequence that proceeds with a common ratio normally denoted by r .In general a geometric sequence is a sequence of the form a,ar,ar^2,ar^3,...,ar^(n-1),where a is the first term, r is the common ratio,n is the number of terms in the sequence and ar^(n-1) is the last term or the nth term of the sequence. For example; 2,4,8,16,32,... is a geometric sequence with first term a=2, and common ratio r=2

A geometric series or the geometric progression is the sum of terms of a geometric sequence i.e. a+ar+ar^2+ar^3+...+ar^(n-1) e.g. 3+9+27+81+...is an infinite geometric series with first term a=3 and common ratio r=3.