Sequence and series are one of the basic topics in Arithmetic. Here, you will learn the Sequence and Series | Progression | Type of Sequence.
A Sequence is an arrangement of numbers in a definite order according to some rule.
Example: 5, 5.5, 5.55, 5.555—————etc.
The various numbers that occur in the sequence are known as its terms. The terms are denoted by a1, a2, a3 — or t1, t2, t3 —
The nth term is the number at the nth position of the sequence and as tn or an.
We define a sequence is non-empty set x to be
map f: n-> x or a map f: N-> x
If X = R is called a real sequence.
If X = C is called a complex sequence.
A sequence that contains the infinite number of terms is known as an infinite sequence. In such sequence no terms is uncountable.
Example: a1, a2, a3, a4, a5, ———-
A sequence that contains the finite number of terms is known as an finite sequence. In such sequence no terms is countable.
Example: a1, a2, a3 —an
Types of Sequence
Most common examples of sequences are:
1. Arithmetic Sequences: If the difference between two consecutive terms is a constant is called Arithmetic Sequences.
Example: 2, 4, 6, 8, 10……….
2. Geometric Sequences: A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence.
Example: 2, 4, 8, 16, 32, 64………..
3. Harmonic Sequences: A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence.
Example: The sequence 1,2,3,4…. is an arithmetic progression, so its reciprocals 11, 12, 13, 14….. are a harmonic progression.
4. Fibonacci Sequence A sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2
Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …
Those sequences which follow certain patterns are generally referred to as progression.
Example: 2, 6, 10, 14 is a progression.
Series is the sum of sequences of terms. The series is finite or infinite according to the given sequence is finite or infinite. Series are represented as sigma, which indicates that the summation.
For example, “2, 4 6, 8” is a sequence, with terms “2”, “4”, “6”, and “8”; a series S can be,
S = Sum (2, 4, 6, 8) = 20
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