Mathematics covers arithmetic series and sequences as a few among the basic topics. While a sequence could be expressed as a collection of elements which allows repetition of any sorts; a series is a sum of all the elements. Solving problems and understanding its fundamentals is likely to make the concept clearer. These are however similar to sets, the primary difference lies in the fact those similar elements can repeatedly occur while dealing with sequences. Sequences could be finite or infinite and the length of terms depends on that.
Definition
A “sequence” could be referred as an ordered list of numbers and is often addresses as a progression. The numbers included in this ordered list are called “terms” or “elements” of the sequence. In a sequence, the set of numbers or objects are known to be set in an order that certainly follows some specific rule. For instance, if x1, x2, x3, x4, ………etc comprise the elements of a sequence, then here, 1, 2, 3, 4,… represents the position of those elements. Based on the number of terms a sequence can be termed as finite or infinite.
On the other hand, a “series” could be expressed as the product of all the terms or elements of the sequence. Also, the resulting value is called the “summation” or “sum”. For instance, if x1, x2, x3, x4,….. is a given sequence then the series for it would be determined as: SN = x1 + x2 + x3 + x4 +…..+ xN . The nature of the sequence, determines the nature of series being finite or infinite.
Types of Series and Sequence
Mentioned below are some of the most common types of sequences:
- Arithmetic Sequence: It is the one where every element is known to be created by either subtracting or adding a definite number to the preceding term in the considerable arithmetic sequence or progression.
- Geometric Sequence: A sequence where every element is known to be obtained by either dividing or multiplying a definite number to the preceding element is referred as a geometric sequence.
- Harmonic Sequence: Any given series of numbers form a harmonic sequence if the reciprocals of all the elements in that sequence form an arithmetic sequence.
- Fibonacci Numbers: These numbers tend to form a special kind of sequence where all elements are known to be obtained by adding up two preceding sequence Also, these sequences start with 0 and 1. A sequence of Fibonacci numbers would be defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2…
These basic details will help you understand sequences and series better.