Over many years, the issue of finding the n th term of real sequences given some terms of the sequence is quite a herculean task and is in no way easy to come by. Obtaining the terms of a real sequence when the n th term is known, is a mathematical science whereas finding the n th term of a sequence given some terms of a sequence is both a mathematical innovation and creative process. On the subject of sequence, numbers of textbooks are worth mentioning for example, Dawikins and Paul, Hazewinkel and Michiel, Howard et al. Abolbashari and Aghaeinia, Benward and Saker, Caplin, Guu-Chang, Jue et al. As for as the application of sequences is concerned, we refer some other applications in music and communication technology e.g. For some well-known examples, we mention here the exponential series, Taylor’s series, Maclaurin’s series, and Fourier’s series. The terms of the series are often produced according to a certain rule, such as by a formula, or by an algorithm. An example is the famous series from Zeno's dichotomy and its mathematical representation: These can be written more compactly using the summation symbol ∑. Given an infinite sequence of numbers, a series is informally the result of adding all those terms together: a 1 + a 2 + a 3 +…. Real sequences have vital role to generate formula and series in real analysis. What will be the total number of seats in the theatre? Write the sequence for the number of seats for the first 5 rows The number of seats in each row can be modelled by the formula C( n) = 16 + 4 n, when n refers to the n th row, and you need 50 rows of seats. Suppose you want to construct a movie theatre in your town. Here is a stunning example to introduce the topic. Sequences and patterns arise naturally in many real life situations. Mathematical sequences can be used to model real life applications. We might identify a n = f( n) for all n or just write a n : N → R. ![]() In other words, a sequence is a map f( n) : N → R. In real analysis a sequence is a function from a subset of the natural numbers to the real numbers. Whether dull or masterly, however, the emphasis is on the underlying process rather than the material itself.īasically, sequences are countably many numbers arranged in an ordered set that may or may not exhibit certain patterns.Ī sequence is usually defined as a function whose domain is a countable totally ordered set, although in many disciplines the domain is restricted, such as to the natural numbers. It is particularly prevalent in passages involving extension or elaboration indeed, because of its inherently directed nature, it was (and still is) often pulled from the shelf by the less imaginative tonal composer as the stock response to a need for transitional or developmental activity. The device of sequence epitomizes both the goal-directed and the hierarchical nature of common-practice tonality. Although stereotypically associated with Baroque music, and especially the music of Antonio Vivaldi, this device is widespread throughout Western music history. The non-diatonic sequence tends to modulate to a new tonality or to cause temporarily tonicisation.Īt least two instances of a sequential pattern including the original statement are required to identify a sequence, and the pattern should be based on several melody notes or at least two successive harmonies (chords). A modulating sequence is a sequence that leads from one tonal center to the next, with each segment technically being in a different key in some sequences.Ī sequence can be described according to its direction (ascending or descending in pitch) and its adherence to the diatonic scale that is, the sequence is diatonic if the pitches remain within the scale, or chromatic (or non-diatonic) if pitches outside of the diatonic scale are used and especially if all pitches are shifted by exactly the same interval (i.e., they are transposed). ![]() A false sequence is a literal repetition of the beginning of a figure and stating the rest in sequence, we refer Benward and Saker. A modified sequence is a sequence where the subsequent segments are decorated or embellished so as to not destroy the character of the original segment. A tonal sequence is a sequence where the subsequent segments are diatonic transpositions of the first segments. ![]() A real sequence is a sequence where the subsequent segments are exact transpositions of the first segment. Sequences have an ancient history dating back at least as far as Archimedes who used sequences and series in his “Method of Exhaustion" to compute better values of ¼ and areas of geometry.
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