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Recurrent sequences
Mina Šekularac
Regional Talent Centre, Belgrade II, minasekularac@hotmail.com
1. Introduction
the reproduction of living organisms. Fibonacci was one of
This paper deals with recurrent sequences. The sequences the most accomplished mathematicians of the Middle Ages.
in which a general term of the sequence depends on its
predecessors are recurrent sequences. The concept in Arithmetic and geometric sequences
mathematics and programming in which the same function
is repeatedly applied in order to solve a problem is known The sequence of real numbers, where each consecutive
as recursion. This mathematical process has been used since member is found by adding the fixed number to the
XII century in order to provide solutions to various laws of previous member, is called arithmetic sequence. The term
science. is the difference of the sequence. The arithmetic
sequence corresponds to the relation:
, for (1)
2. Working methods The sequence where each member, starting from the second
one, is found by multiplying the previous one by the fixed
The following research methods are to be applied in order number , is called geometrical sequence. The term is
to carry out the recurrent sequence research: the quotient. Geometrical sequence corresponds to the
data gathering and analysis using the literature on this recurrent formula:
topic, , for (2)
analysis of previous researches in this field,
previous data processing and systematization,
conclusion methods based on previous facts on general Linear differential equation
information.
Relation of the form:
Research results regarding the recurrent sequences and their
application in the field of mathematics, programming and (3)
everyday life, will be summerized and presented as the
mixture of previous information and the latest conclusions for (k is a constant and ) is called linear
based on this research. differential (recurrent) equation of order k.
Characteristic equation of the differential equation of order
k is the equation of order k:
3. Rеsearch results (4)
Sequences
4. Conclusion
If any natural number , in accordance with some law of
science, corresponds to a real number , it is a never- The researches on recurrent sequences have been conducted
ending sequence of real numbers . The for centuries. In addition to theoretical and practical
numbers are the members of the sequence significance in the field of mathematics and computational
science, certain chemical reactions are carried out in
th
and is member of general member of the sequence accordance with the principle of recurrent sequences. When
. it comes to Nature, the process of reproduction and the
Better known sequences: number of individuals are based on the principles of
1° – the sequence of natural numbers, recurrent sequence. Solving various IT problems is usually
is a general member of the sequence; performed by continual repetition of specific methods
2° - the sequence of even (functions). The application of recurrent functions has
numbers, is a general member of the sequence; affected the development of computational technologies
3° - the sequence of prime and led to improvement of modern technological
numbers; achievements.
4° - harmonious sequence, is a general
member of the sequence; Literature
5° - The Fibonacci Sequence;
6° ... [1.] Z. Kadelburg, V. Mićić, S. Ognjanović: Analiza sa
algebrom 3 (Analysis and algebra), Krug Beograd,
If the sequence has the limiting value x, the sequence
is convergent and it converges to x. If the sequence 2011.
does not have a limiting value, then the sequence [2.] L. Stefanović, M. Matejić, S. Marinković:
diverges and it is divergent. Diferencijalne jednačine (Differential equations) , SKC
Niš, 2006.
Historical review of recurrent sequences [3.] L. Stefanović: Teorija nizova (Theory of Sequences),
SKC Niš, 2010.
th
The term recurrent equation was first mentioned in the 12
century, in theoretical works of Leonardo Pisano, known as [4.] Murraz Spiegel: Calculus of Finite Differences and
Difference Equations, Schaum, 1971.
Fibonacci. He was exploring numerous scientific laws on
