[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-588853-105":59,"doc-detail-588853-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","teaching-recurrent-sequences-in-brazil-using-historical-facts-and-graphical-illustrations","TEACHING RECURRENT SEQUENCES IN BRAZIL USING HISTORICAL FACTS AND GRAPHICAL ILLUSTRATIONS","","The work proposes a mathematics study focused on teaching the history of linear and recurrent second-order sequences, highlighted through Fibonacci, Lucas, Pell, Jacobsthal, Leonardo, Oresme, Mersenne, Padovan, Perrin, and Narayana. Beginning with Fibonacci, the study traces how other sequences emerge, sometimes derived from Fibonacci and sometimes obtained by changing the order. Recurrence formulas and characteristic equations are analyzed to examine their roots and connections to number values such as gold and related proportions, presenting results from Brazil–Portugal research cooperation.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/teaching-recurrent-sequences-in-brazil-using-historical-facts-and-graphical-illustrations/588853/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/teaching-recurrent-sequences-in-brazil-using-historical-facts-and-graphical-illustrations/588853.png","ImageObject",300,407,{"name":92,"@type":93},"Chumphorn","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-15",true,{"@type":102,"interactionType":103,"userInteractionCount":29},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"Which sequences are emphasized in the proposal for teaching recurrent sequences?","Question",{"text":112,"@type":113},"The proposal highlights Fibonacci, Lucas, Pell, Jacobsthal, Leonardo, Oresme, Mersenne, Padovan, Perrin, and Narayana, within the context of teaching linear and recurrent second-order sequences.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the study explain the relationship among these sequences, starting from Fibonacci?",{"text":117,"@type":113},"It notes that other sequences can emerge from Fibonacci—either being derived from it or obtained by changing the order—leading to a historical and mathematically rigorous exploration.",{"name":119,"@type":110,"acceptedAnswer":120},"What mathematical tools does the work use to analyze these recurrent sequences?",{"text":121,"@type":113},"It checks recurrence formulas and characteristic equations, then examines their roots and their relationship to well-known numerical proportions such as gold, silver, bronze, and others.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},588853,1792209086,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":29,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145,"preview_page_count":4},2336475401981,"https://ap-avatar.wpscdn.com/avatar/22000c94efd8d5204d?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786935347598174694","TEACHING RECURRENT SEQUENCES IN BRAZIL USING HISTORICAL FACTS AND GRAPHICAL ILLUSTRATIONS  \nFrancisco Regis Vieira ALVES, Paula Maria Machado Cruz CATARINO, Renata Passos Machado VIEIRA, Milena Carolina dos Santos MANGUEIRA  \nAbstract: The present work presents a proposal for study and investigation, in the context of the teaching of Mathematics, through the history of linear and recurrent 2nd order sequences, indicated by: Fibonacci, Lucas, Pell, Jacobsthal, Leonardo, Oresme, Mersenne, Padovan, Perrin and Narayana. Undoubtedly, starting from the Fibonacci sequence, representing the most popular in the referred scenario of numerical sequences, it was possible to notice the emergence of other sequences. In some cases, being derived from Fibonacci, in others, changing the order of the sequence, we thus have the historical study, with the use of mathematical rigor. Besides, their respective recurrence formulas and characteristic equations are checked, observing their roots and the relationship with the number of gold, silver, bronze and others. The results indicated represent an example of international research cooperation involving researchers from Brazil and Portugal.  \nKey words: Historical-mathematical investigation, Recurring Sequences, History of Mathematics.  \n1. Introduction  \nLinear and recurring sequences have many applications in several areas, such as Science, Arts, Mathematical Computing and others, thus showing an increasing interest of scholars in this subject. The Fibonacci sequence is the best known and most commented on by authors of books on the History of Mathematics, being, therefore, an object of study for many years, until today, safeguarding a relevant character (Rosa, 2012) . Originated from the problem of immortal rabbits, this sequence is shown in many works succinctly and trivially, not providing the reader with an understanding of how we can obtain other sequences, from the evolution of Fibonacci sequence. (Alves, 2017) .  \nRepresenting an inexhaustible source of many interesting algebraic identities, the Fibonacci numbers are presented in the form of a recurring sequence, as then being one of the most famous numerical sequences in mathematics and a popular topic for mathematical enrichment. Thus, this sequence appears in numerous mathematical problems and other ludic problems, highlighting a numerical text in which Fibonacci does an important job for the theory of numbers and solutions of algebraic equations (Posamentier & Lehmann, 2007) .  \nTherefore, it is possible to notice the appearance of several recurring linear sequences, as a result of the mobilization of mathematical scholars from all over the world, with the aim of providing new ways of generalizing it. In view of this, this work has the interest of discussing the historical and epistemological process of evolution of linear and recurrent sequences, through a historical and mathematical bias (Plofker, 2009) .  \nThe deep interest of this research developed from a scientific cooperation involving Brazil and Portugal, was seen in view of works referring to these sequences, in which they treat their respective mathematical, historical and evolutionary processes individually and quickly, with a greater emphasis in the area of Pure Mathematics. On the other hand, it is very important for the students to understand the diversity of applications and examples in the environment, in which we can identify certain notions intrinsically related to these abstract entities. With this, a scientific dissemination of the recurring sequences known to date has been structured. however, disregarded by the authors of History of  \nMathematics books used in Brazil, with attention intermediating their historical and epistemological process, in addition to comparing their respective recurrence formula with Fibonacci numbers. Starting immediately, we have a brief description of what is a linear and recurring sequence, for a posteriori to develop a historical and ma","cbCailOEexkOxGZ0","https://ap.wps.com/l/cbCailOEexkOxGZ0","pdf",659252,18,"English","# 1. Introduction\n# 2. Recurring linear sequence\n## 2.1. The Fibonacci sequence","[{\"question\":\"Which sequences are emphasized in the proposal for teaching recurrent sequences?\",\"answer\":\"The proposal highlights Fibonacci, Lucas, Pell, Jacobsthal, Leonardo, Oresme, Mersenne, Padovan, Perrin, and Narayana, within the context of teaching linear and recurrent second-order sequences.\"},{\"question\":\"How does the study explain the relationship among these sequences, starting from Fibonacci?\",\"answer\":\"It notes that other sequences can emerge from Fibonacci—either being derived from it or obtained by changing the order—leading to a historical and mathematically rigorous exploration.\"},{\"question\":\"What mathematical tools does the work use to analyze these recurrent sequences?\",\"answer\":\"It checks recurrence formulas and characteristic equations, then examines their roots and their relationship to well-known numerical proportions such as gold, silver, bronze, and others.\"}]","TEACHING RECURRENT SEQUENCES IN BRAZIL USING HISTORICAL FACTS AND GRAPHICAL ILLUSTRATIONS | PDF",1792066062,45]