{"title":"由黄金比例导出的斐波那契和卢卡斯恒等式","authors":"K. Adegoke","doi":"10.47443/ejm.2022.018","DOIUrl":null,"url":null,"abstract":"By expressing Fibonacci and Lucas numbers in terms of the powers of the golden ratio α = (1 + √ 5) / 2 and its inverse β = − 1 /α = (1 − √ 5) / 2 , a multitude of Fibonacci and Lucas identities have been developed in the literature. In this paper, the reverse course is followed: numerous Fibonacci and Lucas identities are derived by making use of the well-known expressions for the powers of α and β in terms of Fibonacci and Lucas numbers.","PeriodicalId":29770,"journal":{"name":"International Electronic Journal of Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Fibonacci and Lucas Identities Derived via the Golden Ratio\",\"authors\":\"K. Adegoke\",\"doi\":\"10.47443/ejm.2022.018\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"By expressing Fibonacci and Lucas numbers in terms of the powers of the golden ratio α = (1 + √ 5) / 2 and its inverse β = − 1 /α = (1 − √ 5) / 2 , a multitude of Fibonacci and Lucas identities have been developed in the literature. In this paper, the reverse course is followed: numerous Fibonacci and Lucas identities are derived by making use of the well-known expressions for the powers of α and β in terms of Fibonacci and Lucas numbers.\",\"PeriodicalId\":29770,\"journal\":{\"name\":\"International Electronic Journal of Mathematics Education\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2022-08-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Electronic Journal of Mathematics Education\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.47443/ejm.2022.018\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"EDUCATION & EDUCATIONAL RESEARCH\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Electronic Journal of Mathematics Education","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.47443/ejm.2022.018","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"EDUCATION & EDUCATIONAL RESEARCH","Score":null,"Total":0}
Fibonacci and Lucas Identities Derived via the Golden Ratio
By expressing Fibonacci and Lucas numbers in terms of the powers of the golden ratio α = (1 + √ 5) / 2 and its inverse β = − 1 /α = (1 − √ 5) / 2 , a multitude of Fibonacci and Lucas identities have been developed in the literature. In this paper, the reverse course is followed: numerous Fibonacci and Lucas identities are derived by making use of the well-known expressions for the powers of α and β in terms of Fibonacci and Lucas numbers.
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