{"id":535,"date":"2012-12-27T01:47:57","date_gmt":"2012-12-26T16:47:57","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=535"},"modified":"2021-10-31T07:01:15","modified_gmt":"2021-10-30T22:01:15","slug":"tkb201102","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkb201102\/","title":{"rendered":"\u6771\u5927\u6587\u7cfb2011\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u5b9f\u6570 \\(x\\) \u306e\u5c0f\u6570\u90e8\u5206\u3092, \\(0 \\leqq y \\lt 1\\) \u304b\u3064 \\(x-y\\) \u304c\u6574\u6570\u3068\u306a\u308b\u5b9f\u6570 \\(y\\) \u306e\u3053\u3068\u3068\u3057, \u3053\u308c\u3092\u8a18\u53f7 \\(\\langle x \\rangle\\) \u3067\u8868\u3059.\r\n\u5b9f\u6570 \\(a\\) \u306b\u5bfe\u3057\u3066, \u7121\u9650\u6570\u5217 \\(\\{ a _ n \\}\\) \u306e\u5404\u9805 \\(a _ n \\quad ( n = 1 , 2 , 3 , \\cdots )\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u9806\u6b21\u5b9a\u3081\u308b.<\/p>\r\n

    \r\n

  1. (i)<\/strong>\u3000\\(a _ 1 = \\langle a \\rangle\\)<\/p><\/li>\r\n

  2. (ii)<\/strong>\u3000\\(\\left\\{ \\begin{array}{ll} a _ {n+1} = \\left\\langle \\dfrac{1}{a _ n} \\right\\rangle & \\left( \\ a _ n \\neq 0 \\text{\u306e\u3068\u304d} \\ \\right) \\\\ \\ a _ {n+1} = 0 & \\left( \\ a _ n = 0 \\text{\u306e\u3068\u304d} \\ \\right) \\end{array} \\right.\\)<\/p><\/li>\r\n<\/ol>\r\n

      \r\n

    1. (1)<\/strong>\u3000\\(a = \\sqrt{2}\\) \u306e\u3068\u304d, \u6570\u5217 \\(\\{ a _ n \\}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

    2. (2)<\/strong>\u3000\u4efb\u610f\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066 \\(a _ n = a\\) \u3068\u306a\u308b\u3088\u3046\u306a \\(\\dfrac{1}{3}\\) \u4ee5\u4e0a\u306e\u5b9f\u6570 \\(a\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

      \r\n\r\n

      \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

      \\(1 \\lt a \\lt 2\\) \u306a\u306e\u3067\r\n\\[\r\na _ 1 = \\sqrt{2} -1\r\n\\]\r\n\\(a _ k = \\sqrt{2} -1 \\quad ( k = 1 , 2 , \\cdots )\\) \u3068\u4eee\u5b9a\u3059\u308b\u3068, \\(\\dfrac{1}{\\sqrt{2} -1} = \\sqrt{2} +1\\) \u3067\u3042\u308a, \\(2 \\lt \\sqrt{2} +1 \\lt 3\\) \u306a\u306e\u3067\r\n\\[\r\na _ {k+1} = \\left( \\sqrt{2} +1 \\right) -2 = \\sqrt{2} -1\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066 \\(n \\geqq 1\\) \u306e\u3068\u304d\r\n\\[\r\na _ n = \\underline{\\sqrt{2} -1}\r\n\\]\r\n

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

      \u6761\u4ef6\u3092\u307f\u305f\u3059 \\(a\\) \u306f, \u3042\u308b\u8ca0\u3067\u306a\u3044\u6574\u6570 \\(r\\) \u3092\u7528\u3044\u305f\u4ee5\u4e0b\u306e\u5f0f\u3092\u6e80\u305f\u3059.\r\n\\[\\begin{align}\r\n\\dfrac{1}{a} & = a+r \\\\\r\n\\text{\u2234} \\quad a^2 +ra -1 & = 0 \\quad ... [1]\r\n\\end{align}\\]\r\n\\(\\langle a \\rangle\\) \u3092\u4e0e\u3048\u308b\u5f0f\u304a\u3088\u3073 \\(a \\geqq \\dfrac{1}{3}\\) \u3088\u308a\r\n\\[\\begin{align}\r\n\\dfrac{1}{3} & \\leqq a \\lt 1 \\\\\r\n\\text{\u2234} \\quad 1 & \\lt \\dfrac{1}{a} \\leqq 3\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\u6574\u6570 \\(r\\) \u306e\u5019\u88dc\u306f, \\(r = 0 , 1 , 2\\) \u306e\u307f.
      \r\n\u305d\u308c\u305e\u308c\u306e\u5834\u5408\u306b\u3064\u3044\u3066, [1] \u3088\u308a<\/p>\r\n

        \r\n

      1. 1*<\/strong>\u3000\\(r=0\\) \u306e\u3068\u304d\r\n\\[\r\na^2 -1 = 0 \\\\\r\n\\text{\u2234} \\quad a = 1\r\n\\]\r\n\u306a\u306e\u3067\u4e0d\u9069<\/p><\/li>\r\n

      2. 2*<\/strong>\u3000\\(r=1\\) \u306e\u3068\u304d\r\n\\[\r\na^2 +a -1 = 0 \\\\\r\n\\text{\u2234} \\quad a = \\dfrac{\\sqrt{5} -1}{2}\r\n\\]<\/li>\r\n

      3. 3*<\/strong>\u3000\\(r=2\\) \u306e\u3068\u304d\r\n\\[\r\na^2 +2a -1 = 0 \\\\\r\n\\text{\u2234} \\quad a = \\sqrt{2} -1\r\n\\]<\/li>\r\n<\/ol>\r\n

        \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b \\(a\\) \u306e\u5024\u306f\r\n\\[\r\na = \\underline{\\dfrac{\\sqrt{5} -1}{2} , \\sqrt{2} -1}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5b9f\u6570 \\(x\\) \u306e\u5c0f\u6570\u90e8\u5206\u3092, \\(0 \\leqq y \\lt 1\\) \u304b\u3064 \\(x-y\\) \u304c\u6574\u6570\u3068\u306a\u308b\u5b9f\u6570 \\(y\\) \u306e\u3053\u3068\u3068\u3057, \u3053\u308c\u3092\u8a18\u53f7 \\(\\langle x \\rangle\\) \u3067\u8868\u3059. \u5b9f\u6570 \\(a\\ … \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":[79],"tags":[149,13],"class_list":["post-535","post","type-post","status-publish","format-standard","hentry","category-tokyo_b_2011","tag-tokyo_b","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/535","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=535"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/535\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=535"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=535"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=535"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}