{"id":1298,"date":"2015-10-11T21:32:39","date_gmt":"2015-10-11T12:32:39","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1298"},"modified":"2021-09-29T22:42:42","modified_gmt":"2021-09-29T13:42:42","slug":"tbr201503","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tbr201503\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2015\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(p\\) \u3068 \\(q\\) \u306f\u6b63\u306e\u6574\u6570\u3068\u3059\u308b.\r\n\\(2\\) \u6b21\u65b9\u7a0b\u5f0f \\(x^2 -2px -q = 0\\) \u306e \\(2\\) \u3064\u306e\u5b9f\u6570\u89e3\u3092 \\(\\alpha , \\beta\\) \u3068\u3059\u308b. \u305f\u3060\u3057, \\(\\alpha \\gt \\beta\\) \u3068\u3059\u308b. \u6570\u5217 \\(\\{ a _ n \\}\\) \u3092\r\n\\[\r\na _ n = \\dfrac{1}{2} \\left( {\\alpha}^{n-1} +{\\beta}^{n-1} \\right) \\quad ( n = 1, 2, 3, \\cdots )\r\n\\]\r\n\u306b\u3088\u3063\u3066\u5b9a\u3081\u308b. \u305f\u3060\u3057, \\({\\alpha}^0 = 1\\) , \\({\\beta}^0 = 1\\) \u3068\u5b9a\u3081\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \\(a _ {n+2} = 2p a _ {n+1} +q a _ n\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \\(a _ n\\) \u306f\u6574\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057, \\(\\dfrac{{\\alpha}^{n-1}}{2}\\) \u4ee5\u4e0b\u306e\u6700\u5927\u306e\u6574\u6570\u3092 \\(b _ n\\) \u3068\u3059\u308b. \\(p\\) \u3068 \\(q\\) \u304c \\(q \\lt 2p+1\\) \u3092\u6e80\u305f\u3059\u3068\u304d, \\(b _ n\\) \u3092 \\(a _ n\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\n\\alpha +\\beta = 2p , \\ \\alpha \\beta = -q \\quad ... [1]\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n{\\alpha}^{n+1} +{\\beta}^{n+1} & = ( \\alpha +\\beta ) ( {\\alpha}^n +{\\beta}^n ) -\\alpha \\beta ( {\\alpha}^{n-1} +{\\beta}^{n-1} ) \\\\\r\n& = 2p ( {\\alpha}^n +{\\beta}^n ) +q ( {\\alpha}^{n-1} +{\\beta}^{n-1} )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u4e21\u8fba\u3092 \\(\\dfrac{1}{2}\\) \u500d\u3059\u308c\u3070\r\n\\[\r\na _ {n+2} = 2p a _ {n+1} +q a _ n\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    \u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066, \u300c \\(a _ n\\) \u306f\u6574\u6570\u3067\u3042\u308b. \u300d... [A] \u3053\u3068\u3092, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(n = 1 , 2\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\na _ 1 & = \\dfrac{1}{2} ( 1+1 ) = 1 , \\\\\r\na _ 2 & = \\dfrac{1}{2} ( \\alpha +\\beta ) = p\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u3069\u3061\u3089\u3082 [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(n = k , k+1 \\ ( k \\geqq 1 )\\) \u306e\u3068\u304d\u306b, [A] \u304c\u6210\u7acb\u3059\u308b, \u3059\u306a\u308f\u3061, \\(a _ k , a _ {k+1}\\) \u304c\u3068\u3082\u306b\u6574\u6570\u3067\u3042\u308b\u3068\u4eee\u5b9a\u3059\u308b.
      \r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(a _ {k+2}\\) \u3082\u6574\u6570\u3068\u306a\u308a, \\(n = k+2\\) \u306e\u3068\u304d\u306b [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

      (3)<\/strong><\/p>\r\n

      \u6761\u4ef6\u3088\u308a\r\n\\[\r\n\\alpha = p +\\sqrt{p^2 +q} , \\ \\beta = p -\\sqrt{p^2 +q}\r\n\\]\r\n\\(q \\lt 2p+1\\) \u3088\u308a, \\(p^2 +q \\lt (p+1)^2\\) \u306a\u306e\u3067\r\n\\[\r\np \\lt \\sqrt{p^2 +q} \\lt p+1\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n2p \\lt \\alpha \\lt 2p+1 , \\ -1 \\lt \\beta \\lt 0 \\quad ... [2]\r\n\\]\r\n\u6761\u4ef6\u3088\u308a\r\n\\[\r\n\\dfrac{{\\alpha}^{n-1}}{2} = a _ n -\\underline{\\dfrac{{\\beta}^{n-1}}{2}} _ {[3]}\r\n\\]\r\n(2)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(a _ n\\) \u306f\u6574\u6570\u3067\u3042\u308a, \u307e\u305f [2] \u3088\u308a\u4e0b\u7dda\u90e8 [3] \u306b\u3064\u3044\u3066\r\n\\[\r\n\\left\\{ \\begin{array}{ll} 0 \\lt [3] \\lt \\dfrac{1}{2} & ( \\ n \\ \\text{\u304c\u5947\u6570\u306e\u3068\u304d} \\ ) \\\\ -\\dfrac{1}{2} \\lt [3] \\lt 0 & ( \\ n \\ \\text{\u304c\u5076\u6570\u306e\u3068\u304d} \\ ) \\end{array} \\right.\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nb _ n = \\underline{\\left\\{ \\begin{array}{ll} a _ n -1 & ( \\ n \\ \\text{\u304c\u5947\u6570\u306e\u3068\u304d} \\ ) \\\\ a _ n & ( \\ n \\ \\text{\u304c\u5076\u6570\u306e\u3068\u304d} \\ ) \\end{array} \\right.}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(p\\) \u3068 \\(q\\) \u306f\u6b63\u306e\u6574\u6570\u3068\u3059\u308b. \\(2\\) \u6b21\u65b9\u7a0b\u5f0f \\(x^2 -2px -q = 0\\) \u306e \\(2\\) \u3064\u306e\u5b9f\u6570\u89e3\u3092 \\(\\alpha , \\beta\\) \u3068\u3059\u308b. \u305f\u3060\u3057, \\(\\alpha \\g … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[132],"tags":[144,137],"class_list":["post-1298","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2015","tag-tsukuba_r","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1298","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=1298"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1298\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1298"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1298"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1298"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}