廣義fibonacci數(shù)列相差6、7項(xiàng)的前n項(xiàng)的和式論-文庫(kù)吧
2024-12-31 03:49 本頁(yè)面
【正文】 ugh the nature of the sequence and the general expression formula, We can solve the summation of sequence. In this passage, by using the mathematical elementary method, the generalized sequence difference 6 and 7 of the first n item type have been proved. Thus, the sum of the first n item difference 6 and 7 of Fibonacci sequence、Lucas sequence can be obtained and its numerical value can be easily calculated by the general terms of these sequences.Key Words: General term；Generalized Fibonacci sequence；Fibonacci sequence；Lucas sequence；The sun of the first n item1引言與定義十三世紀(jì)初意大利比薩的一位叫倫納德,綽號(hào)為斐波那契(Fibonacci)的數(shù)學(xué)家[1],提出了一個(gè)有趣的關(guān)于兔子繁殖的問(wèn)題:兔子出生以后兩個(gè)月就能生小兔,若每月不多不少恰好生一對(duì)(一雌一雄).假如養(yǎng)了初生的小兔一對(duì),試問(wèn)一年以后共可有多少對(duì)兔子(假如不發(fā)生死亡)?通過(guò)推算,我們不難得出下面結(jié)果月份12345678910111213兔子數(shù)(對(duì))1123581321345589144233從表中可知:一年后共有兔子233對(duì).我們把上表中下面一列數(shù)用表示,記:則它們被稱(chēng)為斐波那契數(shù)列[2].人們對(duì)這個(gè)數(shù)列的研究興趣歷時(shí)幾百年,得到廣義Fibonacci數(shù)列[2,3],關(guān)于廣義Fibonacci數(shù)列,已有不少的研究成果.廣義Fibonacci數(shù)列可以用如下遞推公式表示[7]:當(dāng)時(shí),就是Fibonacci數(shù)列[8],記為。當(dāng)時(shí), 就是盧卡斯(Lucas)數(shù)列[9],記為.特別地令.文獻(xiàn)[10][11]用數(shù)學(xué)歸納法和初等方法證明了廣義Fibonacci數(shù)列相差小于6項(xiàng)的前ｎ項(xiàng)和式,本文繼續(xù)用初等方法證明它的相差６項(xiàng)、７項(xiàng)的前ｎ項(xiàng)的和式
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