Arithmetic Progressions Research Articles

Fluency and Flexibility of Students in Solving Arithmetic Sequence Problems Based on Self Efficacy

This research is a descriptive qualitative research that aims to describe students' creative thinking in solving arithmetic sequence problems in terms of self-efficacy. This research was conducted at SMK Kesehatan Dhermajaya. The subjects of this study were 3 grade XI students who were taken according to the aspects of self-efficacy, namely magnitude, strength, generality. Creative thinking indicators in this study are fluency, flexibility, and novelty. The results of this study indicate that the magnitude subject in the fluency aspect is that students can provide 2 possible jars needed. The flexibility aspect is that students can solve with 2 alternative solutions with the arithmetic sequence formula, and write an arithmetic sequence that is at odds with 2 and if added together the result is 90. The strength of the fluency aspect is that students provide 2 possible jars needed. At the same time, the aspects of flexibility and novelty have not been fulfilled. The subject of generality in the fluency aspect is that students can provide 3 possible jars. The flexibility aspect is that students can provide 2 different alternative solutions, namely using the arithmetic sequence formula and writing an arithmetic sequence that is 2 apart and then adding them up. The three subjects, still have not shown the uniqueness of the answer or used different strategies so that the novelty aspect has not been fulfilled.

Students' Specializing Thinking in Solving Arithmetic Sequence and Series Problems

Specializing thinking is a mathematical thinking process that is very important in the mathematics learning process. Specializing thinking is thinking by starting with special things. The application of specializing thinking can be an effective strategy for teachers to improve students' mathematical thinking abilities in dealing with various problems. This research aims to describe students' specializing forms of thinking in solving problems of arithmetic sequences and series. This research method is descriptive qualitative research with a case study approach. The subjects in this research were 2 class X high school students who successfully solved arithmetic sequence and series problems. The instruments used are tests and interviews. The data in this research are the results of the subject's work and the results of interview transcripts. The research results show that two forms of specializing thinking were found in solving arithmetic sequences and series problems, that is explicit schematic representation and implicit schematic representation. It is hoped that the results of this research can provide a valuable contribution to mathematics teachers in designing more meaningful learning.

Exploration of mathematical concepts in prayer and Mushofahah movements

Knowledge of integration between mathematics and religion is necessary to increase a Muslim's mastery of scientific treasures. This article describes several mathematical concepts contained in prayer movements and Mushofahah. This research is a qualitative descriptive research type of literature review. Data/information collection methods are carried out by observation and documentation. These two methods are used to collect information about mathematical concepts contained in prayer movements and Mushofahah, obtained from several primary and secondary sources in the form of the al-Qur’an, al-Hadith, books, and scientific journals as well as documentation of prayer movements and Mushofahah (shaking hands). The data analysis technique used is content analysis. The research results showed many daily life activities, including prayer and Mushofahah, concrete examples of the integration of mathematics and Islamic religion. These concepts include: In the prayer movement, there were number concepts, which included the addition and multiplication of integers, algebraic concepts in the form of sets, arithmetic sequences, and geometric concepts in the form of lines and angles. Meanwhile, in Mushofahah, there were factorial concepts, permutation concepts, and opportunities.

On roots of quadratic congruences

AbstractThe equidistribution of roots of quadratic congruences with prime moduli depends crucially upon effective bounds for special Weyl linear forms. Duke, Friedlander and Iwaniec discovered strong estimates for these Weyl linear forms when the quadratic polynomial has negative discriminant. Tóth proved analogous but weaker bounds when the quadratic polynomial has positive discriminant. We establish strong estimates for these Weyl linear forms for quadratics of positive discriminants. As an application of our bounds, we derive a quantitative uniform distribution of modular square roots with integer moduli in an arithmetic progression.

Families of numerical semigroups and a special case of the Huneke-Wiegand conjecture

The Huneke-Wiegand conjecture is a decades-long open question in commutative algebra. García-Sánchez and Leamer showed that a special case of this conjecture concerning numerical semigroup rings k [ Γ ] can be answered in the affirmative by locating certain arithmetic sequences within the numerical semigroup Γ. In this paper, we use their approach to prove the Huneke-Wiegand conjecture in the case where Γ is generated by a generalized arithmetic sequence and showcase how visualizations can be leveraged to find the requisite arithmetic sequences.

Etnomatematika: Eksplorasi Amak Lampisan Mandailing dalam Pembelajaran Matematika

The decline in cultural understanding among society is something that must be taken seriously and requires ongoing study, because culture is one of the distinctive characteristics of a nation. Linking the surrounding culture with learning is one step to preserve and maintain this culture for future generations. Amak lampisan Mandailing is a typical woven mat from the Mandailing Tribe which has elements of mathematical concepts in it. The ethnomathematics studied are mathematical concepts related to learning material at SMP/MTs and equivalent schools. This research uses exploratory research with an ethnographic approach to study the relationship between culture and learning materials. explore the shapes and patterns of amaklampisan. The aim of this research is to bring out cultural wisdom in mathematics learning so that it can motivate and influence students' learning achievement. This type of research is exploratory qualitative research about Amak Lampisan Mandailing. Based on the results of this research, Mandailing lampisan amak is related to material in mathematics such as rectangles, congruence and similarity, arithmetic sequences and series, social arithmetic.

Application of the Self Organized Learning Environment Learning Model in Mathematics Learning Arithmetic Sequence Material

The objective of this study is to analyze the disparities in the mean learning achievements between students who utilize the SOLE learning model and those who employ the direct learning model for arithmetic sequence material at Tondano State Vocational School. The research employed a quasi-experimental design with a Posttest Only Control Group Design. The participants in this study consisted of students from class X OTKP 1, who were assigned to the experimental group, and students from class X OTKP 2, who were assigned to the control group. The data gathering method involved administering a technical learning outcomes test at the conclusion of the class (Posttest). The data analysis technique employed in this research utilizes tests on data that has undergone normality and homogeneity testing. The t test calculations yielded a t_count value of 2.736, which exceeded the critical t_table value of 1.685. Therefore, the null hypothesis (H_0) was rejected, and the alternative hypothesis (H_1) was accepted. Therefore, it can be inferred that there is a disparity in the mean learning results of students in the Self-Organized Learning Environment (SOLE) instructional approach and the traditional direct learning approach when it comes to arithmetic sequence material.

The distributions of coefficients of triple product $L$-functions over arithmetic progressions

The distributions of coefficients of triple product $L$-functions over arithmetic progressions

Triangles with arithmetic, geometric and harmonic progressions

This article explores the fascinating world of triangles with angle and side sequences in arithmetic, geometric and harmonic progressions. The properties and relationships between these progressions and the corresponding triangle parameters are investigated. The main focus is on proving a series of theorems that provide insights into the conditions under which triangles exhibit these progressions. The proofs are presented with meticulous attention to detail, offering a deep understanding of the results. The presented findings contribute to the field of mathematical education by providing rich material for classroom activities and discussions.

Fast Multivariate Multipoint Evaluation over All Finite Fields

Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field \(\mathbb {F}\) that outputs the evaluations of an m -variate polynomial of degree less than d in each variable at N points in time, \(\begin{equation*} (d^m+N)^{1+o(1)}\cdot {{\sf poly}}(m,d,\log |\mathbb {F}|), \end{equation*}\) for all \(m\in \mathbb {N}\) and all sufficiently large \(d\in \mathbb {N}\) . A previous work of Kedlaya and Umans (FOCS 2008 and SICOMP 2011) achieved the same time complexity when the number of variables m is at most \(d^{o(1)}\) and had left the problem of removing this condition as an open problem. A recent work of Bhargava, Ghosh, Kumar, and Mohapatra (STOC 2022) answered this question when the underlying field is not too large and has characteristic less than \(d^{o(1)}\) . In this work, we remove this constraint on the number of variables over all finite fields, thereby answering the question of Kedlaya and Umans over all finite fields. Our algorithm relies on a non-trivial combination of ideas from three seemingly different previously known algorithms for multivariate multipoint evaluation, namely the algorithms of Kedlaya and Umans, that of Björklund, Kaski, and Williams (IPEC 2017 and Algorithmica 2019), and that of Bhargava, Ghosh, Kumar, and Mohapatra, together with a result of Bombieri and Vinogradov from analytic number theory about the distribution of primes in an arithmetic progression. We also present a second algorithm for multivariate multipoint evaluation that is completely elementary and, in particular, avoids the use of the Bombieri–Vinogradov theorem. However, it requires a mild assumption that the field size is bounded by an exponential tower in d of bounded height . More specifically, our second algorithm solves the multivariate multipoint evaluation problem over a finite field \(\mathbb {F}\) in time, \(\begin{equation*} (d^m+N)^{1+o(1)}\cdot {{\sf poly}}(m,d,\log |\mathbb {F}|), \end{equation*}\) for all \(m\in \mathbb {N}\) and all sufficiently large \(d\in \mathbb {N}\) , provided that the size of the finite field \(\mathbb {F}\) is at most \((\exp (\exp (\exp (\cdots (\exp (d)))))\) , where the height of this tower of exponentials is fixed.

Arithmetic progressions in polynomial orbits

Let f be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit [Formula: see text], where t is an integer, using arithmetic progressions each of which contains t. Fixing an integer [Formula: see text], we prove that it is impossible to cover [Formula: see text] using k such arithmetic progressions unless [Formula: see text] is contained in one of these progressions. In fact, we show that the relative density of terms covered by k such arithmetic progressions in [Formula: see text] is uniformly bounded from above by a bound that depends solely on k. In addition, the latter relative density can be made as close as desired to 1 by an appropriate choice of k arithmetic progressions containing t if k is allowed to be large enough.

Cyclicity and Exponent of Elliptic Curves Modulo p in Arithmetic Progressions

Abstract In this article, we study the cyclicity problem of elliptic curves $E/\mathbb{Q}$ modulo primes in a given arithmetic progression. We extend the recent work of Akbal and Güloğlu by proving an unconditional asymptotic for such a cyclicity problem over arithmetic progressions for elliptic curves E, which also presents a generalization of the previous works of Akbary, Cojocaru, M.R. Murty, V.K. Murty and Serre. In addition, we refine the conditional estimates of Akbal and Güloğlu, which gives log-power savings (for small moduli) and consequently improves the work of Cojocaru and M.R. Murty. Moreover, we study the average exponent of E modulo primes in a given arithmetic progression and obtain several conditional and unconditional estimates, extending the previous works of Freiberg, Kim, Kurlberg and Wu.

A note on odd partition numbers

Ramanujan’s partition congruences modulo ℓ∈{5,7,11}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell \\in \\{5, 7, 11\\}$$\\end{document} assert that p(ℓn+δℓ)≡0(modℓ),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} p(\\ell n+\\delta _{\\ell })\\equiv 0\\pmod {\\ell }, \\end{aligned}$$\\end{document}where 0<δℓ<ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\delta _{\\ell }<\\ell $$\\end{document} satisfies 24δℓ≡1(modℓ).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$24\\delta _{\\ell }\\equiv 1\\pmod {\\ell }.$$\\end{document} By proving Subbarao’s conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes ℓ≥5,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell \\ge 5,$$\\end{document} we give a new proof of the conclusion that there are infinitely many m for which p(ℓm+δℓ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(\\ell m+\\delta _{\\ell })$$\\end{document} is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest such m satisfies m<(ℓ2-1)/24,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m<(\\ell ^2-1)/24,$$\\end{document} representing a significant improvement to the previous bound.

Learning the weights using attribute order information for multi-criteria decision making tasks

In multi-criteria decision making, the importance of decision criteria (decision attributes) plays a crucial role. Ranking is a useful technique for expressing the importance of decision criteria in a decision-makers’ preference system. Since weights are commonly utilized for characterizing the importance of criteria, weight determination and assessment are important tasks in multi-criteria decision making and in voting systems as well. In this study, we concentrate on the connection between the preference order of decision criteria and the decision weights. Here, we present an easy-to-use procedure that can be used to produce a sequence of weights corresponding to a decision-makers’ preference order of decision criteria. The proposed method does not require pairwise comparisons, which is an advantageous property especially in cases where the number of criteria is large. This method is based on the application of a class of regular increasing monotone quantifiers, which we refer to as the class of weighting generator functions. We will show that the derivatives of these functions can be used for approximating the criteria weights. Also, we will demonstrate that using weighting generator functions, weights can be inverted in a consistent way. We will deduce the generators for arithmetic and geometric weight sequences, and we will present a one-parameter generator function known as the tau function in continuous-valued logic. We will show that using these weighting generator functions, the weight learning task can be turned into a simple, one-parameter optimization problem.

Simulation of a multi-level absorption thermal battery with variable solution flow rate for adjustable cooling capacity

Absorption thermal battery (ATB) has been regarded as a promising solution to increase the penetration of renewable energy in the energy supply. Currently, one of the biggest challenges for the ATB application is to overcome the attenuation problem in the discharging process caused by solution dilution. This study proposes a novel multi-level ATB to achieve adjustable cooling capacities. A three-level ATB is performed as an example and compared with the conventional ATB. Moreover, the solution flow rate regulation is further applied to reach more accurate supply-demand matching. Then, the effects of various parameters on the discharging performance are investigated. Results indicate that instead of gradually attenuating from 17.2 kW to 4.3 kW of the conventional ATB, three stable and switchable cooling capacities (i.e., 13.7 kW, 8.1 kW, and 5.1 kW) are achieved by a three-level ATB. The energy storage density of the three-level ATB (123.2 kWh/m3) is also higher than that of the conventional ATB (114.5 kWh/m3). Besides, three stable loads distributed in an arithmetic series have been satisfied accurately by the three-level absorption thermal battery with solution flow rate regulation. This work aims to provide theoretical references and suggestions for the development and applications of ATBs.

Growing patterns invention by primary education students

ABSTRACT This paper addresses the creation of patterns through an open-ended invention task. With a qualitative study, we aimed to characterise the written answers of 76 primary education students in fourth, fifth and sixth grades, identifying the type of pattern they invented. We analysed the representations they used and the structure followed in the sequences they created. In addition, we determined whether there were differences in the invention of patterns among the three school grades in the sample. Results showed that students prefer creating growing patterns using numerical and visual representations. They generally involved a numerical structure more frequently than a geometric one, using an increasing arithmetic progression. They usually formed the pattern by involving a geometric structure, building a figure based on an initial element to which they added more elements to complete a particular figure. In fourth grade, numerical representations were more common than pictorial ones, while in sixth grade, the opposite was true. As age increased, so did the frequency with which students invented a pattern. Results also provided information on the complexity of the types of geometric patterns invented.

Layers of Mathematical Understanding and Folding Back of Prospective Teachers on Arithmetic Sequence and Series

Mathematical understanding is one of the objectives of learning mathematics, so students need to master it. The success of the process of growing students' mathematical understanding is inseparable from the role of the teacher. Therefore, student teacher candidates must have a complete mathematical understanding structure. This is what prompted him to carry out research to describe the layers of mathematical understanding and folding back prospective teacher students on the material of arithmetic sequences and series. The researcher took 18 sixth-semester students of S1 Mathematics Education at the State University of Malang. A purposive sampling technique was used to determine the participants. So, one subject was chosen in each category of mathematical understanding. Researchers used a descriptive approach with three stages of research procedures including planning, implementing, and drawing conclusions. The instruments used consisted of test questions and interview guidelines. Based on analysis carried out using indicators for each layer of mathematical understanding, information was obtained that subjects with high abilities were able to achieve layering arrangement and perform effective folding back independently. Subjects with moderate abilities were able to reach the observation layer and fold back effectively but needed help. Meanwhile, subjects with low abilities were able to reach the formalization layer but had not been able to fold back effectively. It can be concluded that subjects with better mathematical understanding have a more complete layered structure of mathematical understanding and can fold back independently and effectively.

PENGEMBANGAN MODUL BERBASIS BRAIN BASED LEARNING PADA POKOK BAHASAN BARISAN DAN DERET ARITMATIKA TERHADAP PEMAHAMAN KONSEP SISWA

This study aims to develop a Brain Based Learning – based module on the subject of arithmetic sequences and series onstudents’ conceptual understanding of SMP Negeri 1 Amarasi Timur. This research was conducted at Amarasi Timur 1 Publik Middle School, Odd Semester, 2022/2023 Academic Year. The type of research used is research and development (R&amp;D), namely the development of teaching tools including teaching material modules, test questions, and student response questionnaires. The development method for this device is using a 4D model which includes: (1) the definition stage (Define), (2) the design stage (Design), (3) the development stage (Develop), and (4) the deployment stage (Dessiminate). The subjects in this research trial were class VIII students of SMPN 1 Amarasi Timur and to find out the response of teachers and students to the development of learning modules based on brain based learning. Data analysis techniques in this study were validation tests carried out by media experts and material experts as well as practicality and effectiveness. The results of the study showed that the validation results of Brain Based Learning modules by media experts were 3,58 in the Very Valid category, and results of material experts were 3,50 in the very valid category, overall the average module validation was 3,54 in the very valid category. The feasibility results of the learning module are based on the results of the assessment that the learning module achieves an average total score of 3,06 out of a maximum score of 4% in the practical category. This can be interpreted that media experts state that the learning module is in the valid category an is suitable for use in learning.

From discrete to continuous: Monochromatic 3-term arithmetic progressions

We prove a known 2-coloring of the integers [ N ] ≔ { 1 , 2 , 3 , … , N } [N] ≔\{1,2,3,\dots ,N\} minimizes the number of monochromatic arithmetic 3-progressions under certain restrictions. A monochromatic arithmetic progression is a set of equally-spaced integers that are all the same color. Previous work by Parrilo, Robertson and Saracino conjectured an optimal coloring for large N N that involves 12 12 colored blocks. Here, we prove that the conjectured coloring minimizes monochromatic arithmetic 3 3 -progressions among anti-symmetric colorings with 12 12 or fewer colored blocks. We leverage a connection to the coloring of the continuous interval [ 0 , 1 ] [0,1] used by Parrilo, Robertson, and Saracino as well as by Butler, Costello and Graham. Our proof identifies classes of colorings with permutations, then counts the permutations using mixed integer linear programming.

On the least almost-prime in arithmetic progressions

On the least almost-prime in arithmetic progressions