If we have a sequence as x1, x2 ,x3,......., and the sequence follows the rule as xj - xi = d for all values of i and j, then the sequence is called the arithmetic sequence. This type of sequence where any term except the first term is obtained by adding a fixed number to the previous term.
A sequence where any term except the first term is obtained by multiplying a fixed number to the previous term is called geometric sequence. If the sequence x1, x2, x3,.......,xn follows the condition xn+1xnxn+1xn = r, then that sequence is known as geometric sequence and r is known as the common ration of the sequence.
This is a type of sequence where the reciprocal of the terms from an arithmetic sequence. So, if any sequence is in the form of 1x1,1x2,1x3,.......,1xn1x1,1x2,1x3,.......,1xn, then the sequence is said to be harmonic sequence.If a, b and c are in the form of arithmetic sequence, then b = 2aca+c2aca+cFor example, the sequence 11,12,13,.....11,12,13,..... is a harmonic sequence.
In some cases, an arrangement of numbers such as 1, 1, 2, 3, 5, 8,.. has no visible pattern. But the sequence is generated by the recurrence relation and is given bya1 = a2 = 1a3 = a1 + a2an = an - 2 + an - 1, n > 2This type of sequence is called Fibonacci sequence.