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It targets notable definition features and addresses limited attention in the History of Mathematics literature, especially regarding geometric representation. The study follows a bibliographical, exploratory methodology and produces geometric constructions of Fibonacci, Lucas, Jacobsthal, and Padovan identities. GeoGebra is shown as essential for interpreting recurrence relations and sequence properties via their visual behavior.",{"@graph":69,"@context":114},[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/recurrent-and-linear-sequences-a-three-dimensional-visualization-with-support-of-geogebra-software/588265/",{"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/recurrent-and-linear-sequences-a-three-dimensional-visualization-with-support-of-geogebra-software/588265.png","ImageObject",300,407,{"name":92,"@type":93},"Aditya","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-15",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108],{"name":109,"@type":110,"acceptedAnswer":111},"What research methodology is used?","Question",{"text":112,"@type":113},"The study uses a bibliographical, exploratory methodology with theoretical support from prior works cited in the text.","Answer","https://schema.org",{"og:url":83,"og:type":116,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":118,"canonical":83},"index,follow",{"doc_id":120,"site_id":62},588265,1792189059,{"code":4,"msg":5,"data":123},{"doc_id":120,"user_id":124,"nickname":92,"user_avatar":125,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":126,"file_id":127,"file_url":128,"file_type":129,"file_size":130,"view_count":34,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":131,"language":132,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":133,"faqs":134,"seo_title":135,"seo_description":67,"update_tm":136,"read_time":36,"preview_page_count":4},962085564549,"https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","RECURRENT AND LINEAR SEQUENCES: A THREEDIMENSIONAL VISUALIZATION WITH SUPPORT OF  \nGEOGEBRA SOFTWARE  \nFrancisco Régis Vieira ALVES, Carla Patrícia Souza Rodrigues PINHEIRO, Renata Teófilo DE  \nSOUSA, Paula Maria Machado Cruz CATARINO  \nAbstract: This article is an excerpt from a master's thesis developed in Brazil, in which we approach recurrent and linear sequences, given some intriguing particularities in their definitions and the scarcity of discussion of this topic in the literature of the History of Mathematics, especially with regard to its geometric representation. Thus, we aim to present the identities of Fibonacci, Lucas, Jacobsthal and Padovan in a three-dimensional visualization with the contribution of GeoGebra software. The research methodology chosen was bibliographical, exploratory in nature, where we have theoretical support in works such as Oliveira and Alves (2019), Silva (2017), Souza and Alves (2018), Vieira and Alves (2020) . This research brings as results a set of geometric constructions of the identities of the proposed sequences, in three-dimensional perspective, being a support for future works developed around this theme. GeoGebra was essential in the process of constructing and visualizing the sequences, as it provided strategies for understanding the recurrence relations and the properties of the Fibonacci, Lucas, Jacobsthal and Padovan sequences, through the behavior of the visual representations of these identities.  \nKey words: Recurrent and linear sequences; GeoGebra; Geometric visualization.  \n1. Introduction  \nThe recurrent and linear sequences are explored in several areas such as Arts, Computing, Science, Mathematics, among others, which denotes a high interest of the scientific community for this subject. The Fibonacci sequence is the most explored by authors of the History of Mathematics, due to its eternalization with the problem of immortal rabbits, which is explored several times in a brief and trivial way. According to Alves (2017), readers cannot trivially understand the evolution of other sequences through the Fibonacci sequence, even though this sequence is the basis for building the others.  \nThe Fibonacci numbers, through unlimited sources, are represented by several algebraic identities, becoming an important sequence for the development of Mathematics (Alves et al., 2020) . Faced with research such as Oliveira and Alves (2019), Silva (2017), Souza and Alves (2018), Vieira and Alves (2020), which address studies on the generalization of Fibonacci, Lucas, Jacobsthal and Padovan sequences, respectively, there was a need to explore the geometric form of identities related to these sequences, a non-trivial component that was still little explored in previous research. Thus, the objective of this work is to present the identities of Fibonacci, Lucas, Jacobsthal and Padovan in three-dimensional visualization with the contribution of the GeoGebra software.  \nAlves (2022a) reinforces the importance of some elements for mathematical learning regarding recurrent and linear sequences, which are “the ability to visualize and identify geometric patterns, the adoption of a mathematical notational system consistent with algebraic, arithmetic and geometric invariants involved in each problem and the ability to understand and solve problems”(p. 322) .  \nOther studies such as Alves and Catarino (2017), Mangueira (2022) and Vieira et al. (2019) also point out the lack of exploration of these sequences in the context of the History of Mathematics within undergraduate degrees in Brazil, both with regard to mathematical discussion and their exploration through technological resources. Alves (2022a, 2022b) points out that currently, the History of  \nMathematics is still studied separately from Mathematics itself, disregarding it as the “conductor thread”of an entire epistemological and evolutionary process. Thus, these authors bring research that explore the discussion of these recursive linear seque","cbCaichYd5DvRJdC","https://ap.wps.com/l/cbCaichYd5DvRJdC","pdf",549559,16,"English","# Introduction\n## Recursive and linear sequences","[{\"question\":\"What research methodology is used?\",\"answer\":\"The study uses a bibliographical, exploratory methodology with theoretical support from prior works cited in the text.\"}]","RECURRENT AND LINEAR SEQUENCES: A THREE-DIMENSIONAL VISUALIZATION WITH SUPPORT OF GEOGEBRA SOFTWARE | PDF",1792064872]