![]() ![]() Sequences (1) and (3) are examples of divergent sequences. Sequences that are not convergent are said to be divergent. For example, sequences (2) and (4) are convergent, and their limits are 0 and the function 1/(1 + x 2), respectively. The following points are helpful to clearly understand the concepts of sequence and series. The limit of a sequence of functions is defined in a similar manner. If the terms of a sequence of numbers differ by an arbitrarily small amount from the number a for sufficiently large n, the sequence is said to be convergent, and a is called its limit. Since the common difference is latex8 /latex or written as latexd8 /latex, we can find the next term after latex31 /latex by adding latex8 /latex. First, find the common difference of each pair of consecutive numbers. Example 1: Find the next term in the sequence below. The sequences most often encountered are those of numbers or functions. Examples of How to Apply the Concept of Arithmetic Sequence. ![]() Fibonacci numbers, for example, are defined through a recurrence formula. Here, the constant number is called a common difference, represented by d. In an arithmetic sequence, each term is obtained by adding a constant number to the previous term (except the first term). To define a sequence, we can either specify its nth term or make use of a recurrence formula, by which each term is defined as a function of preceding terms. An arithmetic sequence is a series of numbers related to each other by a constant addition or subtraction. Different terms of a sequence may be identical.Ī sequence may be regarded as a function whose argument can take on only positive integral values-that is, a function defined on the set of natural numbers. ![]() The elements of which it is composed are called its terms. The Fibonacci sequence of numbers, say Fn where the suffix n denotes the order or rank of term, is defined by. It can be written in the form x 1, x 2, …, x n, … or simply. A sequence is a set of elements of any nature that are ordered as are the natural numbers 1,2,…, n…. ![]()
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