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APPLICATION OF NUMERICAL SEQUENCES IN ECONOMICS.
Iqbaljon Khaydarov
Kokand University, Teacher, Department of Digital
Technologies and Mathematics.
ANNOTATION:
This article focuses on the use of numerical sequences in economics. Analysis
of economic processes and their planning requires the use of mathematical tools. Also, many
people have heard of arithmetic progression, but not everyone can interpret it in economics.
Arithmetic progression is used to model economic processes that have a tendency to constantly
increase and decrease. Not only arithmetic progression, but also geometric progression can be
used in economics. The geometric progression section also presents methods for finding the
terms of the geometric progression studied in the section and methods for finding the sum of the
terms of their formulas. The importance of numerical sequences, progressions and their
microeconomic analysis, as well as operations in business planning, is also covered. Numerical
sequences are an important tool for understanding the dynamics of economic growth and
effective management. The application of numerical sequences to economics can be expressed in
the following way: there are formulas for finding interest rates on loans from banks and loan
amortizations. Two different methods are used to calculate these interest rates. They are
arithmetic progression for simple interest, and geometric progression for compound interest. This
process can be called debt amortization in short. In debt amortization, it can also be used to find
the percentage of the remaining balance of a loan from a bank, variable interest rates, and the
amount of interest that remains after the loan is paid.
Keywords:
numbers, percentages, geometric progression, arithmetic sequence, formula,
example, microeconomic analysis.
INTRODUCTION
Today, great attention is paid to the field of mathematics not only in our country, but also in
foreign countries. Currently, there are several presidential decrees and laws aimed at developing
and supporting mathematics. In particular, the presidential decree “On measures to improve the
quality of education and develop scientific research in the field of mathematics” was adopted. At
the same time, from September 1, 2021, it is mandatory for mathematics teachers in specialized
schools to have a national certificate of the appropriate level. In 2020-2022, work was carried out
to develop a program of measures for training highly qualified personnel in mathematics for
economic sectors and the social sphere. In accordance with the Resolution of the President of the
Republic of Uzbekistan No. PQ-4708 dated May 7, 2020, the financing of the activities of the
Laboratory "Coordination of Educational and Methodological Materials for Mathematics
Education" is carried out from the funds of the State Budget of the Republic of Uzbekistan
within the funds allocated for the maintenance of the Institute. A laboratory for the coordination
of teaching and methodological materials for mathematics education, consisting of 5 staff units,
was established within the Institute. The Ministry of Finance of the Republic of Uzbekistan, the
Ministry of Innovative Development and the Academy of Sciences approved the proposal of the
President of the Republic of Uzbekistan to establish the Muhammad al-Khwarizmi International
Prize in order to reward scientists who, based on the results of fundamental and applied scientific
research in the field of mathematics, have proposed a solution to a specific problem in practice.
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In Uzbekistan, the introduction of modern pedagogical technologies for the formation of early
mathematical ideas in preschool children, the development of specialized schools in the regions,
and the establishment of new schools are considered a priority. In accordance with this, the
President of the Republic of Uzbekistan adopted a resolution “On measures to improve the
quality of education and develop scientific research in the field of mathematics”. In accordance
with this resolution, by November 1, 2020, specialized schools for in-depth teaching of
mathematics (Specialized Schools) will be gradually established in each district (city).
According to the presidential decree, in order to improve the quality of education in mathematics
in Uzbekistan, specialized schools for mathematics will be established in each district. A national
certification system for assessing the level of knowledge of mathematics has been introduced.
An international award named after Muhammad Al-Khwarizmi has also been established, and its
winners will be awarded 50 thousand dollars.
Methodology
There are three types of numerical sequences. They are divided into arithmetic, geometric and
arbitrary progressions. An arbitrary sequence, a sequence based on a certain rule, is called an
arbitrary sequence. A sequence formed by adding the same number is called an arithmetic
sequence. When finding the last progression, that is, a geometric progression, it is called a
progression formed by multiplying the same number. When finding the interest rate on a loan
from a bank, these progressions are used depending on simple or compound interest. In
particular, if the interest on the money received is in simple interest, it can be calculated
arithmetically, and if it is in compound interest, it can be calculated in geometric progression.
Many people have not yet come across the concept of interest. Interest is the fee we use. Debt
amortization is the repayment of a loan taken in several periods at equal intervals. Financial rent
is of two types. The principal payment of the loan and the interest payment for the remaining
balance.
1. Arithmetic sequence:
In an arithmetic sequence, each subsequent element is formed by adding or subtracting a certain
number from the previous one. For example, here each number is increased by 3 from the
previous one. The general formula for such a sequence is:
1.
a
n
= a
1
+(n-1)d Finding the nth term of an arithmetic progression
2.
d=a
2
-a
1
d ni topish
3.
Finding the solution of all terms
4.
Finding the solution of all terms
5.
find the median of a
2. Optional sequence:
This sequence is a special sequence in which each element is equal to the sum of the previous
two elements. Numerical sequences are the main tool for solving various mathematical problems.
Their various forms and areas of application are multifaceted and are important in the
development of mathematical thinking. Understanding and applying the properties of each type
of sequence in practice helps to achieve success in any field. The following inequality is proved
for arbitrary natural numbers n:
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In arithmetic progression 1) if a
1
=2,d=3 if a
15
= find the. If a
1
=-3
d= -2 ifa
18
= find the a
n
=a
1
+(n-1)*d
Solving: a
1
=2 d=3 a
15
=2+(15-1)*3=2+42=44
a
1
= -3 d= -2 a
18
= -3+(18-1)*(-2)= -35
Depreciable value is the sum of the initial (replacement) cost of an asset as stated in the financial
statements, less the expected (estimated) residual value. For fixed assets, the initial cost is the
depreciable value increased by the amount of costs for additional construction, provision of
additional equipment, reconstruction, modernization, technical re-equipment, and after the
completion of these works, the residual (balance sheet) value of these fixed assets, determined at
the time of their commissioning, minus the expected (estimated) residual value. Depreciation is a
value expression of depreciation in the form of a systematic distribution and transfer of the
depreciable value of an asset over its useful life to the cost of products (works, services) or
period costs, based on the function of the fixed assets; For example, we also use a numerical
sequence to draw up a debt amortization schedule. Example 1. At the beginning of the year,
Ahmed borrowed 40,000 soums from a bank at an annual interest rate of 50%. Determine the
interest payments on the debt and the debt amortization for each quarter?
Stages
of
Debt
Repayment
(t)
Loan Amount
(K)
Interest Rate
(R%)
Interest
Payment
Amount
(I
t)
Payments
on
the
Principal
Loan
Amount
(K
0
)
Quarterly
Contribution
(K
0
+I)
1
40000
12.5%
5000
10000
15000
2
30000
12.5%
3750
10000
13750
3
20000
12.5%
2500
10000
12500
4
10000
12.5%
1250
10000
11250
Jami
-
-
12500
40000
52500
A company's revenue, profit, and expenses are plotted as a series of numbers over time. This data
is analyzed in depth to determine their growth dynamics and make strategic decisions. If a
company's revenue increases by 105% year over year, this forms a geometric progression. This
information can be used to predict how much the company's revenue will be in future years. The
actions of consumers and producers shape supply and demand. This process is often modeled
through series of numbers. If the demand for a certain product increases year over year, its price
will also increase accordingly. Mathematical models construct supply and demand curves, which
help to understand how the equilibrium price is formed in the market.
Results
There are many formulas used in numerical sequences. For example, one example based on this
formula will be considered: d=a2-a1 Finding d
a1=8 a2=10 d=10-8=2 In this example, the next term is found by adding 2 to the arithmetic
progression. a1=8 a2=10 a3=12 a4=14 a5=16.
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Stages of
Debt
Repayment
(t)
Loan
Amount
(K)
Interest
Rate
(R%)
Interest
Payment
Amount
(I
t)
Payments
on
the
Principal
Loan
Amount
(K
0
)
Quarterly
Contribution
(K
0
+I)
1
60 mln
5%
3mln
15 mln
18mln
2
45 mln
5%
2.25mln
15mln
17.5mln
3
30 mln
5%
1.5 mln
15mln
16.5mln
4
15 mln
5%
0.75 mln
15mln
15.75mln
Conclusion
Number sequences are very important in economics for analyzing data, making predictions, and
making optimal decisions. Banks and other credit institutions accumulate money in their hands
with the condition of paying interest to its owners. A loan is understood as a relationship in
which the funds that are temporarily idle in the hands of their owners are borrowed by others for
a certain period, with the condition of paying interest. The time period over which the interest
rate is calculated is understood as the compound interest period. The compound interest period
can be divided into compound interest integrals. If the initial debt is considered unchanged and
the compound interest is calculated for a specific period, then the simple interest rate is used.
This simple interest rate is calculated using an arithmetic progression, and this type of
calculation is presented in this article. In addition, geometric progression is also used for
compound interest. Today, it was announced that these directions are an effective and clear way
to avoid creating difficulties for bank employees.
References:
1.
1.F.Rajabov and others. “Higher Mathematics”, Tashkent “Uzbekistan” 2007.400b
2.
2.R.Jurakulov, S.Akbarov, D.Toshpolatov, Mathematics textbook Tashkent, 2022
3.
3.Haydarov M.Solving differential-functional equations with homogeneous constant
coefficients using the Bruvy series. Bulletin of the Khorezm Mamun Academy-2-1/2024
4.
4.Semyenov,,Ant’ye i mantissa“ Sbornil zadach c resheniyemi. IPM im. M.B. Keldisha
2015g.
5.
5.M.A. Mirzaahmedov, D.A. Sotiboldiev,,Preparing students for mathematical
olympiads“ Tashkent, “Teacher“ 1993y.
6.
6.N.H. Agakhanov, I.I. Bagdanov "Vserossiyskie olympiady shkolnikov po matematike
1993-2006 g." District and final stage. Moscow .Izdatelstvo MTsNMO 2007 g.
7.
7. Add T. , Andrica D. Problems for Mathematical Contests. - GIL Publishing House,
2003.
8.
8. Khaidarov, I. (2024). EFFECTIVE METHODS IN TEACHING MATHEMATICS
FOR ECONOMIC STUDENTS.KOKAN UNIVERSITY BULLETIN, 10, 35-37.
9.
9. Khaidarov, I. I. (2023). METHODS OF SOLUTION OF PROBLEMS IN THE OUT-
OF-SCHOOL OLYMPIAD IN MATHEMATICS OF SCHOOL STUDENTS. QO ‘KON
UNIVERSITY NEWSLETTER, 31-34.
10.
10. Ilyosjon o‘gli, Kh. I. (2024). APPLICATION OF FUNCTIONS AND GRAPHS TO
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ECONOMICS. University Research Base, 569-574.
11.
11. Haydarova K. THE ROLE OF WOMEN IN MODERN ARTIFICIAL
INTELLIGENCE AND ROBOTICS //International Journal of Artificial Intelligence. – 2025. – T.
1. – No. 3. – P. 716-721.
12.
12. Haydarova K. SOIL NPK SENSOR AND ARDUINO: INTELLIGENT
MONITORING SYSTEM FOR HEALTHY PLANT GROWTH //QOKON UNIVERSITY
NEWSLETTER. – 2024. – T. 13. – P. 390-392.
