{"id":373,"date":"2012-04-14T00:14:05","date_gmt":"2012-04-13T15:14:05","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=373"},"modified":"2021-03-23T17:45:43","modified_gmt":"2021-03-23T08:45:43","slug":"kyr201206","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr201206\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb2012\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\u3055\u3044\u3053\u308d\u3092 \\(n\\) \u56de\u6295\u3052\u3066\u51fa\u305f\u76ee\u3092\u9806\u306b \\(X _ 1 , X _ 2 , \\cdots , X _ n\\) \u3068\u3059\u308b. \u3055\u3089\u306b\r\n\\[\r\nY _ 1 = X _ 1 , \\ Y _ k = X _ k +\\dfrac{1}{Y _ {k-1}} \\quad ( k =2, \\cdots , n )\r\n\\]\r\n\u306b\u3088\u3063\u3066 \\(Y _ 1 , Y _ 2 , \\cdots , Y _ n\\) \u3092\u5b9a\u3081\u308b.\r\n\\[\r\n\\dfrac{1+\\sqrt{3}}{2} \\leqq Y _ n \\leqq 1+\\sqrt{3}\r\n\\]\r\n\u3068\u306a\u308b\u78ba\u7387 \\(p _ n\\) \u3092\u6c42\u3081\u3088.<\/p>\r\n

\r\n\r\n

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

\u6f38\u5316\u5f0f\u3088\u308a, \\(Y _ k \\geqq 1 \\ ( 1 \\leqq k \\leqq n)\\) \u306a\u306e\u3067\r\n\\[\r\n0 \\lt \\dfrac{1}{Y _ k} \\leqq 1\r\n\\]\r\n\\(\\dfrac{1+\\sqrt{3}}{2} = 1.36 \\cdots\\) , \\(1+\\sqrt{3} = 2.73 \\cdots\\) \u306a\u306e\u3067, \\(\\dfrac{1+\\sqrt{3}}{2} \\leqq Y _ n \\leqq 1+\\sqrt{3}\\) \u3068\u306a\u308b\u306e\u306f, \\(X _ n =1 , 2\\) \u306e\u3068\u304d\u306b\u9650\u3089\u308c\u308b.<\/p>\r\n

    \r\n

  1. 1*<\/strong>\u3000\\(X _ n =1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n\\dfrac{\\sqrt{3} -1}{2} & \\leqq \\dfrac{1}{Y _ {n-1}} \\leqq \\color{red}{\\sqrt{3}} \\\\\r\n\\text{\u2234} \\quad 1 & \\leqq Y _ {n-1} \\leqq 1 +\\sqrt{3} \\quad ... [1]\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(Y _ {n-1}\\) \u304c[1]\u306e\u7bc4\u56f2\u306b\u3042\u3063\u3066, \\(X _ n =1\\) \u3068\u306a\u308c\u3070\u3088\u3044.<\/p><\/li>\r\n

  2. 2*<\/strong>\u3000\\(X _ n =2\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n0 & \\lt \\dfrac{1}{Y _ {n-1}} \\leqq \\sqrt{3} -1 \\\\\r\n\\text{\u2234} \\quad & Y _ {n-1} \\geqq \\dfrac{1 +\\sqrt{3}}{2} \\quad ... [2]\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(Y _ {n-1}\\) \u304c[2]\u306e\u7bc4\u56f2\u306b\u3042\u3063\u3066, \\(X _ n =1\\) \u3068\u306a\u308c\u3070\u3088\u3044.<\/p><\/li>\r\n<\/ol>\r\n

    \u3053\u3053\u3067, \\(1 \\leqq Y _ n \\leqq \\dfrac{1 +\\sqrt{3}}{2}\\) \u3068\u306a\u308b\u78ba\u7387\u3092 \\(q _ n\\) , \\(Y _ n \\geqq 1 +\\sqrt{3}\\) \u3068\u306a\u308b\u78ba\u7387\u3092 \\(r _ n\\) \u3068\u304a\u304f.
    \r\n\u6f38\u5316\u5f0f\u3088\u308a, \\(Y _ n\\) \u304c\u7121\u7406\u6570\u306b\u306a\u308b\u3053\u3068\u306f\u306a\u3044\u306e\u3067\r\n\\[\r\np _ n +q _ n +r _ n =1 \\quad ... [3]\r\n\\]\r\n

    1*<\/strong> 2*<\/strong>\u3088\u308a, \\(n \\geqq 2\\) \u306b\u5bfe\u3057\u3066\r\n\\[\\begin{align}\r\np _ n & = \\dfrac{1}{6} \\left( p _ {n-1} +\\color{red}{q _ {n-1}} \\right) +\\dfrac{1}{6} \\left( p _ {n-1} +r _ {n-1} \\right) \\\\\r\n& = \\dfrac{1}{6} p _ {n-1} +\\dfrac{1}{6} \\quad ( \\ \\text{\u2235} \\ [3] \\ ) \\\\\r\n\\text{\u2234} \\quad & p _ n -\\dfrac{1}{5} =\\dfrac{1}{6} \\left( p _ {n-1} -\\dfrac{1}{5} \\right)\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(p _ 1 =\\dfrac{1}{6}\\) \u3092\u7528\u3044\u308c\u3070\r\n\\[\r\np _ n -\\dfrac{1}{5} =\\left( \\dfrac{1}{6} \\right)^{n-1} \\left( \\dfrac{1}{6} -\\dfrac{1}{5} \\right) = \\color{red}{-}\\dfrac{1}{30} \\left( \\dfrac{1}{6} \\right)^{n-1}\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\np _ n =\\underline{\\dfrac{1}{5} \\left\\{ 1 \\color{red}{-}\\left( \\dfrac{1}{6} \\right)^{n} \\right\\}}\r\n\\]\r\n

    \u3010 \u4fee\u6b63\u3057\u307e\u3057\u305f \u3011<\/strong><\/p>\r\n

    \u8aa4\u308a\u304c\u3042\u3063\u305f\u305f\u3081, 2012\/12\/30\u8d64\u5b57\u306e\u90e8\u5206\u4fee\u6b63\u3057\u307e\u3057\u305f. \u3054\u6307\u6458\u304f\u3060\u3055\u3063\u305f\u6e05\u6c34\u69d8, \u3042\u308a\u304c\u3068\u3046\u3054\u3056\u3044\u307e\u3059\uff01<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u3055\u3044\u3053\u308d\u3092 \\(n\\) \u56de\u6295\u3052\u3066\u51fa\u305f\u76ee\u3092\u9806\u306b \\(X _ 1 , X _ 2 , \\cdots , X _ n\\) \u3068\u3059\u308b. \u3055\u3089\u306b \\[ Y _ 1 = X _ 1 , \\ Y _ k = X _ k +\\dfrac{ … \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":[59],"tags":[140,68],"class_list":["post-373","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2012","tag-kyoto_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/373","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=373"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/373\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=373"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=373"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=373"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}