Identifying Figurate Number Patterns

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Identifying Figurate Lesson 7-9 Number Patterns DATE TIME SRB 1 Draw the next three rectangular numbers. 58-61 2 a. Complete the list of the first 10 rectangular numbers. 2, 6, 12, 20, 30, 42 56,EM4_MJ2_G4_U07_L09_001A.ai 72, 90, 110 b. How did you get your answers? Sample answer: I kept multiplying the next 2 counting numbers. I knew 42 = 6 ⁎ 7, so to find the next rectangular number, I multiplied 7 ⁎ 8 = 56. 3 a. Continue the following pattern: 2 = 2 2 + 4 = 6 2 + 4 + 6 = 12 2 + 4 + 6 + 8 = 20 2 + 4 + 6 + 8 + 10 = 30 2 + 4 + 6 + 8 + 10 + 12 = 42 b. Describe the pattern. Sample answer: Each rectangular number is the sum of the even numbers in order, starting with 2. c. What pattern do you notice about the number of addends and the rectangular number? The number of addends is the same as the “number” of the rectangular number. For example, 30 is the fifth rectangular number, and it has 5 addends. d. Describe a similar pattern with square numbers. Sample answer: Each square number is the sum of the odd numbers in order, starting with 1. 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16, and so on. 252 4.OA.5, 4.NBT.6, SMP7, SMP8 Identifying Figurate Lesson 7-9 DATE TIME Number Patterns (continued) Try This 4 Triangular numbers are numbers that are the sum of consecutive counting numbers. For example, the triangular number 3 is the sum of 1 + 2, and the triangular number 6 is the sum of 1 + 2 + 3. a. Continue the patterns in the table below to find a connection between triangular numbers and rectangular numbers. Triangular Number 1 3 6 10 15 Model EM4_MJ2_G4_U07_L09_004A.ai Rectangular Number EM4_MJ2_G4_U07_L09_003A.ai2 6 EM4_MJ2_G4_U07_L09_005A.ai12 EM4_MJ2_G4_U07_L09_006A.ai EM4_MJ2_G4_U07_L09_002A.ai 20 30 Array b. What patterns doEM4_MJ2_G4_U07_L09_007A.ai you notice?EM4_MJ2_G4_U07_L09_008A.ai SampleEM4_MJ2_G4_U07_L09_009A.ai answer:EM4_MJ2_G4_U07_L09_010A.ai Each EM4_MJ2_G4_U07_L09_011A.aitriangular number is always half of the corresponding rectangular number. For example, the fourth triangular number is 10 and the fourth rectangular number is 20. c. The 20th rectangular number is 420. What is the 20th triangular number? 210 Explain how you got your answer. Sample answer: I divided 420 by 2. d. Draw an array to show the rectangular number 20. Divide the array with a line to show the connection between rectangular and triangular numbers. EM4_MJ2_G4_U07_L09_012A.ai 4.OA.5, 4.NBT.6, SMP7, SMP8 253.

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Triangular Numbers /, 3,6, 10, 15, ", Tn,'" »*"
TRIANGULAR NUMBERS V.E. HOGGATT, JR., and IVIARJORIE BICKWELL San Jose State University, San Jose, California 9111112 1. INTRODUCTION To Fibonacci is attributed the arithmetic triangle of odd numbers, in which the nth row has n entries, the cen- ter element is n* for even /?, and the row sum is n3. (See Stanley Bezuszka [11].) FIBONACCI'S TRIANGLE SUMS / 1 =:1 3 3 5 8 = 2s 7 9 11 27 = 33 13 15 17 19 64 = 4$ 21 23 25 27 29 125 = 5s We wish to derive some results here concerning the triangular numbers /, 3,6, 10, 15, ", Tn,'" »*". If one o b - serves how they are defined geometrically, 1 3 6 10 • - one easily sees that (1.1) Tn - 1+2+3 + .- +n = n(n±M and (1.2) • Tn+1 = Tn+(n+1) . By noticing that two adjacent arrays form a square, such as 3 + 6 = 9 '.'.?. we are led to 2 (1.3) n = Tn + Tn„7 , which can be verified using (1.1). This also provides an identity for triangular numbers in terms of subscripts which are also triangular numbers, T =T + T (1-4) n Tn Tn-1 • Since every odd number is the difference of two consecutive squares, it is informative to rewrite Fibonacci's tri- angle of odd numbers: 221 222 TRIANGULAR NUMBERS [OCT. FIBONACCI'S TRIANGLE SUMS f^-O2) Tf-T* (2* -I2) (32-22) Ti-Tf (42-32) (52-42) (62-52) Ti-Tl•2 (72-62) (82-72) (9*-82) (Kp-92) Tl-Tl Upon comparing with the first array, it would appear that the difference of the squares of two consecutive tri- angular numbers is a perfect cube.
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An Heptagonal Numbers
© April 2021| IJIRT | Volume 7 Issue 11 | ISSN: 2349-6002 An Heptagonal Numbers Dr. S. Usha Assistant Professor in Mathematics, Bon Secours Arts and Science College for Women, Mannargudi, Affiliated by Bharathidasan University, Tiruchirappalli, Tamil Nadu Abstract - Two Results of interest (i)There exists an prime factor that divides n is one. First few square free infinite pairs of heptagonal numbers (푯풎 , 푯풌) such that number are 1,2,3,5,6,7,10,11,13,14,15,17…… their ratio is equal to a non –zero square-free integer and (ii)the general form of the rank of square heptagonal Definition(1.5): ퟑ ퟐ풓+ퟏ number (푯 ) is given by m= [(ퟏퟗ + ퟑ√ퟒퟎ) + 풎 ퟐퟎ Square free integer: ퟐ풓+ퟏ (ퟏퟗ − ퟑ√ퟒퟎ) +2], where r = 0,1,2…….relating to A Square – free Integer is an integer which is divisible heptagonal number are presented. A Few Relations by no perfect Square other than 1. That is, its prime among heptagonal and triangular number are given. factorization has exactly one factors for each prime that appears in it. For example Index Terms - Infinite pairs of heptagonal number, the 10 =2.5 is square free, rank of square heptagonal numbers, square-free integer. But 18=2.3.3 is not because 18 is divisible by 9=32 The smallest positive square free numbers are I. PRELIMINARIES 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23……. Definition(1.1): A number is a count or measurement heptagon: Definition(1.6): A heptagon is a seven –sided polygon.
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Elementary School Numbers and Numeration
DOCUMENT RESUME ED 166 042 SE 026 555 TITLE Mathematics for Georgia Schools,' Volume II: Upper Elementary 'Grades. INSTITUTION Georgia State Dept. of Education, Atlanta. Office of Instructional Services. e, PUB DATE 78 NOTE 183p.; For related document, see .SE 026 554 EDRB, PRICE MF-$0.33 HC-$10.03 Plus Postage., DESCRIPTORS *Curriculm; Elementary Educatia; *Elementary School Mathematics; Geometry; *Instruction; Meadurement; Number Concepts; Probability; Problem Solving; Set Thory; Statistics; *Teaching. Guides ABSTRACT1 ' This guide is organized around six concepts: sets, numbers and numeration; operations, their properties and number theory; relations and functions; geometry; measurement; and probability and statistics. Objectives and sample activities are presented for.each concept. Separate sections deal with the processes of problem solving and computation. A section on updating curriculum includes discussion of continuing program improvement, evaluation of pupil progress, and utilization of media. (MP) ti #######*#####*########.#*###*######*****######*########*##########**#### * Reproductions supplied by EDRS are the best that can be made * * from the original document. * *********************************************************************** U S DEPARTMENT OF HEALTH, EDUCATION & WELFARE IS NATIONAL INSTITUTE OF.. EDUCATION THIS DOCUMENT HA4BEEN REPRO- DuCED EXACTLY AS- RECEIVEDFROM THE PERSON OR ORGANIZATIONORIGIN- ATING IT POINTS OF VIEWOR 01NIONS STATED DO NOT NECESSARILYEpRE SENT OFFICIAL NATIONAL INSTITUTEOF TO THE EDUCATION4
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Figurate Numbers
Figurate Numbers by George Jelliss June 2008 with additions November 2008 Visualisation of Numbers The visual representation of the number of elements in a set by an array of small counters or other standard tally marks is still seen in the symbols on dominoes or playing cards, and in Roman numerals. The word "calculus" originally meant a small pebble used to calculate. Bear with me while we begin with a few elementary observations. Any number, n greater than 1, can be represented by a linear arrangement of n counters. The cases of 1 or 0 counters can be regarded as trivial or degenerate linear arrangements. The counters that make up a number m can alternatively be grouped in pairs instead of ones, and we find there are two cases, m = 2.n or 2.n + 1 (where the dot denotes multiplication). Numbers of these two forms are of course known as even and odd respectively. An even number is the sum of two equal numbers, n+n = 2.n. An odd number is the sum of two successive numbers 2.n + 1 = n + (n+1). The even and odd numbers alternate. Figure 1. Representation of numbers by rows of counters, and of even and odd numbers by various, mainly symmetric, formations. The right-angled (L-shaped) formation of the odd numbers is known as a gnomon. These do not of course exhaust the possibilities. 1 2 3 4 5 6 7 8 9 n 2 4 6 8 10 12 14 2.n 1 3 5 7 9 11 13 15 2.n + 1 Triples, Quadruples and Other Forms Generalising the divison into even and odd numbers, the counters making up a number can of course also be grouped in threes or fours or indeed any nonzero number k.
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TRIANGULAR REPUNIT—THERE IS but 1 a Repunit Is Any Integer That
Czechoslovak Mathematical Journal, 60 (135) (2010), 1075–1077 TRIANGULAR REPUNIT—THERE IS BUT 1 John H. Jaroma, Ave Maria (Received June 27, 2009) Abstract. In this paper, we demonstrate that 1 is the only integer that is both triangular and a repunit. Keywords: repunit, triangular number MSC 2010 : 11A99 A repunit is any integer that can be written in decimal notation as a string of 1’s. Examples include 1, 11, 111, 1111, 11111,... Furthermore, if we denote rn to be the nth repunit then it follows that 10n − 1 (1) rn = . 9 Triangular numbers are those integers that can be represented by the number of dots evenly arranged in an equilateral triangle. More specifically, they are the numbers 1, 3, 6, 10, 15,... It follows that the nth triangular number, tn is the sum of the first n consecutive integers beginning with 1. In particular, for n > 1 n(n + 1) (2) tn = . 2 Now, the number 1 is both triangular and a repunit. It is the objective of this note to illustrate that 1 is the only such number. To this end, we state and prove the following lemma. It incorporates a result that [1] asserts had been known by the early Pythagoreans. Appearing in Platonic Questions, Plutarch states that eight times any triangular number plus one is a square. The result is actually necessary and sufficient. We demonstrate it as such here. 1075 Lemma. The integer n is triangular if and only if 8n +1 is a square. Proof. If n = tn, then by (2), n(n + 1) 8 +1=4n2 +4n +1=(2n + 1)2.
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PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 105(119) (2019), 93–100 DOI: https://doi.org/10.2298/PIM1919093L PELL FACTORIANGULAR NUMBERS Florian Luca, Japhet Odjoumani, and Alain Togbé Abstract. We show that the only Pell numbers which are factoriangular are 2, 5 and 12. 1. Introduction Recall that the Pell numbers Pm m> are given by { } 1 αm βm P = − , for all m > 1, m α β − where α =1+ √2 and β =1 √2. The first few Pell numbers are − 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025,... Castillo [3], dubbed a number of the form Ftn = n!+ n(n + 1)/2 for n > 1 a factoriangular number. The first few factoriangular numbers are 2, 5, 12, 34, 135, 741, 5068, 40356, 362925,... Luca and Gómez-Ruiz [8], proved that the only Fibonacci factoriangular numbers are 2, 5 and 34. This settled a conjecture of Castillo from [3]. In this paper, we prove the following related result. Theorem 1.1. The only Pell numbers which are factoriangular are 2, 5 and 12. Our method is similar to the one from [8]. Assuming Pm = Ftn for positive integers m and n, we use a linear forms in p-adic logarithms to find some bounds on m and n. The resulting bounds are large so we run a calculation to reduce the bounds. This computation is highly nontrivial and relates on reducing the diophantine equation Pm = Ftn modulo the primes from a carefully selected finite set of suitable primes. 2010 Mathematics Subject Classification: Primary 11D59, 11D09, 11D7; Secondary 11A55.
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Patterns in Figurate Sequences
Patterns in Figurate Sequences Concepts • Numerical patterns • Figurate numbers: triangular, square, pentagonal, hexagonal, heptagonal, octagonal, etc. • Closed form representation of a number sequence • Function notation and graphing • Discrete and continuous data Materials • Chips, two-color counters, or other manipulatives for modeling patterns • Student activity sheet “Patterns in Figurate Sequences” • TI-73 EXPLORER or TI-83 Plus/SE Introduction Mathematics has been described as the “science of patterns.” Patterns are everywhere and may appear as geometric patterns or numeric patterns or both. Figurate numbers are examples of patterns that are both geometric and numeric since they relate geometric shapes of polygons to numerical patterns. In this activity you will analyze, extend, and describe patterns involving figurate numbers and make connections between numeric and geometric representations of patterns. PTE: Algebra Page 1 © 2003 Teachers Teaching With Technology Patterns in Figurate Sequences Student Activity Sheet 1. Using chips or other manipulatives, analyze the following pattern and extend the pattern pictorially for two more terms. • • • • • • • • • • 2. Write the sequence of numbers that describes the quantity of dots above. 3. Describe this pattern in another way. 4. Extend and describe the following pattern with pictures, words, and numbers. • • • • • • • • • • • • • • 5. Analyze Table 1. Fill in each of the rows of the table. Table 1: Figurate Numbers Figurate 1st 2nd 3rd 4th 5th 6th 7th 8th nth Number Triangular 1 3 6 10 15 21 28 36 n(n+1)/2 Square 1 4 9 16 25 36 49 64 Pentagonal 1 5 12 22 35 51 70 Hexagonal 1 6 15 28 45 66 Heptagonal 1 7 18 34 55 Octagonal 1 8 21 40 Nonagonal 1 9 24 Decagonal 1 10 Undecagonal 1 ..
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Polygonal Numbers 1
Polygonal Numbers 1 Polygonal Numbers By Daniela Betancourt and Timothy Park Project for MA 341 Introduction to Number Theory Boston University Summer Term 2009 Instructor: Kalin Kostadinov 2 Daniela Betancourt and Timothy Park Introduction : Polygonal numbers are number representing dots that are arranged into a geometric figure. Starting from a common point and augmenting outwards, the number of dots utilized increases in successive polygons. As the size of the figure increases, the number of dots used to construct it grows in a common pattern. The most common types of polygonal numbers take the form of triangles and squares because of their basic geometry. Figure 1 illustrates examples of the first four polygonal numbers: the triangle, square, pentagon, and hexagon. Figure 1: http://www.trottermath.net/numthry/polynos.html As seen in the diagram, the geometric figures are formed by augmenting arrays of dots. The progression of the polygons is illustrated with its initial point and successive polygons grown outwards. The basis of polygonal numbers is to view all shapes and sizes of polygons as numerical values. History : The concept of polygonal numbers was first defined by the Greek mathematician Hypsicles in the year 170 BC (Heath 126). Diophantus credits Hypsicles as being the author of the polygonal numbers and is said to have came to the conclusion that the nth a-gon is calculated by 1 the formula /2*n*[2 + (n - 1)(a - 2)]. He used this formula to determine the number of elements in the nth term of a polygon with a sides. Polygonal Numbers 3 Before Hypsicles was acclaimed for defining polygonal numbers, there was evidence that previous Greek mathematicians used such figurate numbers to create their own theories.
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The Mathematical Beauty of Triangular Numbers
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The Square Number by the Approximation Masaki Hisasue
The Australian Journal of Mathematical Analysis and Applications http://ajmaa.org Volume 7, Issue 2, Article 12, pp. 1-4, 2011 THE SQUARE NUMBER BY THE APPROXIMATION MASAKI HISASUE Received 15 August, 2010; accepted 19 November, 2010; published 19 April, 2011. ASAHIKAWA FUJI GIRLS’ HIGH SCHOOL,ASAHIKAWA HANASAKI-CHO 6-3899, HOKKAIDO,JAPAN. [email protected] ABSTRACT. In this paper, we give square numbers by using the solutions of Pell’s equation. Key words and phrases: Diophantine equations, Pell equations. 2000 Mathematics Subject Classification. 11D75. ISSN (electronic): 1449-5910 c 2011 Austral Internet Publishing. All rights reserved. 2 MASAKI HISASUE 1. INTRODUCTION A square number, also called a perfect square, is a figurate number of the form n2, where n is an non-negative integer. The square numbers are 0, 1, 4, 9, 16, 25, 36, 49, ··· . The difference between any perfect square and its predecessor is given by the following identity, n2 − (n − 1)2 = 2n − 1. Also, it is possible to count up square numbers by adding together the last square, the last square’s root, and the current root. Squares of even numbers are even, since (2n)2 = 4n2 and squares of odd numbers are odd, since (2n − 1)2 = 4(n2 − n) + 1. It follows that square roots of even square numbers are even, and square roots of odd square numbers are odd. Let D be a positive integer which is not a perfect square. It is well known that there exist an infinite number of integer solutions of the equation x2 − Dy2 = 1, known as Pell’s equation.
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Some Links of Balancing and Cobalancing Numbers with Pell and Associated Pell Numbers
Bulletin of the Institute of Mathematics Academia Sinica (New Series) Vol. 6 (2011), No. 1, pp. 41-72 SOME LINKS OF BALANCING AND COBALANCING NUMBERS WITH PELL AND ASSOCIATED PELL NUMBERS G. K. PANDA1,a AND PRASANTA KUMAR RAY2,b 1 National Institute of Technology, Rourkela -769 008, Orissa, India. a E-mail: gkpanda nit@rediﬀmail.com 2 College of Arts Science and Technology, Bandomunda, Rourkela -770 032, Orissa, India. b E-mail: [email protected] Abstract Links of balancing and cobalancing numbers with Pell and associated Pell numbers th th are established. It is proved that the n balancing number is product of the n Pell th number and the n associated Pell number. It is further observed that the sequences of balancing and cobalancing numbers are very closely related to the Pell sequence whereas, the sequences of Lucas-balancing and Lucas-cobalancing numbers constitute the associated Pell sequence. The solutions of some Diophantine equations including Pythagorean and Pythagorean-type equations are obtained in terms of these numbers. 1. Introduction The study of number sequences has been a source of attraction to the mathematicians since ancient times. From that time many mathematicians have been focusing their attention on the study of the fascinating triangu- lar numbers (numbers of the form n(n + 1)/2 where n Z+ are known as ∈ triangular numbers). Behera and Panda [1], while studying the Diophan- tine equation 1 + 2 + + (n 1) = (n + 1) + (n +2)+ + (n + r) on · · · − · · · triangular numbers, obtained an interesting relation of the numbers n in Received April 28, 2009 and in revised form September 25, 2009.
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Basic Properties of the Polygonal Sequence
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 26 February 2021 doi:10.20944/preprints202102.0385.v3 Basic Properties of the Polygonal Sequence Kunle Adegoke [email protected] Department of Physics and Engineering Physics, Obafemi Awolowo University, 220005 Ile-Ife, Nigeria 2010 Mathematics Subject Classification: Primary 11B99; Secondary 11H99. Keywords: polygonal number, triangular number, summation identity, generating func- tion, partial sum, recurrence relation. Abstract We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature. 1 Introduction For a fixed integer r ≥ 2, the polygonal sequence (Pn,r) is defined through the recurrence relation, Pn,r = 2Pn−1,r − Pn−2,r + r − 2 (n ≥ 2) , with P0,r = 0,P1,r = 1. Thus, the polygonal sequence is a second order, linear non-homogeneous sequence with constant coefficients. In § 1.2 we shall see that the polygonal numbers can also be defined by a third order homogeneous recurrence relation. The fixed number r is called the order or rank of the polygonal sequence. The nth polygonal number of order r, Pn,r, derives its name from representing a polygon by dots in a plane: r is the number of sides and n is the number of dots on each side. Thus, for a triangular number, r = 3, for a square number, r = 4 (four sides), and so on.
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