{"id":452,"date":"2012-10-29T23:22:18","date_gmt":"2012-10-29T14:22:18","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=452"},"modified":"2021-09-16T06:02:00","modified_gmt":"2021-09-15T21:02:00","slug":"ngr201003","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201003\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2010\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\u306f\u3058\u3081\u306b, A \u304c\u8d64\u7389\u3092 \\(1\\) \u500b, B \u304c\u767d\u7389\u3092 \\(1\\) \u500b, C \u304c\u9752\u7389\u3092 \\(1\\) \u500b\u6301\u3063\u3066\u3044\u308b.\r\n\u8868\u88cf\u306e\u51fa\u308b\u78ba\u7387\u304c\u305d\u308c\u305e\u308c \\(\\dfrac{1}{2}\\) \u306e\u786c\u8ca8\u3092\u6295\u3052, \u8868\u304c\u51fa\u308c\u3070 A \u3068 B \u306e\u7389\u3092\u4ea4\u63db\u3057, \u88cf\u304c\u51fa\u308c\u3070 B \u3068 C \u306e\u7389\u3092\u4ea4\u63db\u3059\u308b, \u3068\u3044\u3046\u64cd\u4f5c\u3092\u8003\u3048\u308b. \u3053\u306e\u64cd\u4f5c\u3092 \\(n\\) \u56de\uff08 \\(n = 1, 2, 3, \\cdots\\) \uff09\u3092\u304f\u308a\u8fd4\u3057\u305f\u5f8c\u306b, A , B , C \u304c\u8d64\u7389\u3092\u6301\u3063\u3066\u3044\u308b\u78ba\u7387\u3092\u305d\u308c\u305e\u308c \\(a _ n , b _ n , c _ n\\) \u3068\u304a\u304f.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a _ 1 , b _ 1 , c _ 1 , a _ 2 , b _ 2 , c _ 2\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(a _ {n+1} , b _ {n+1} , c _ {n+1}\\) \u3092 \\(a _ n , b _ n , c _ n\\) \u3067\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(a _ n , b _ n , c _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n

    \u89e3\u7b54<\/h3>\r\n

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

    \\[\\begin{align}\r\na _ 1 & = \\underline{\\dfrac{1}{2}} , \\ b _ 1 = \\underline{\\dfrac{1}{2}} , \\ a _ 1 = \\underline{0} , \\\\\r\na _ 2 & = \\dfrac{1}{2} \\cdot \\dfrac{1}{2} +\\dfrac{1}{2} \\cdot \\dfrac{1}{2} =\\underline{\\dfrac{1}{2}} , \\\\\r\nb _ 2 & = \\dfrac{1}{2} \\cdot \\dfrac{1}{2} =\\underline{\\dfrac{1}{4}} , \\ c _ 2 =\\dfrac{1}{2} \\cdot \\dfrac{1}{2} =\\underline{\\dfrac{1}{4}}\n\\end{align}\\]\r\n

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

    \\(n\\) \u56de\u76ee\u306e\u5f8c, A , B , C \u304c\u8d64\u7389\u3092\u6301\u3063\u305f\u72b6\u614b\u304b\u3089, \\(n+1\\) \u56de\u76ee\u3067\u8d64\u7389\u3092\u8ab0\u304c\u6301\u3064\u3053\u3068\u306b\u306a\u308b\u304b, \u3092\u8868\u3059\u72b6\u614b\u9077\u79fb\u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n\"\"\r\n

    \u3088\u3063\u3066\r\n\\[\\begin{align}\r\na _ {n+1} & = \\underline{\\dfrac{1}{2} a _ n +\\dfrac{1}{2} b _ n} \\quad ... [1] \\\\\r\nb _ {n+1} & = \\underline{\\dfrac{1}{2} a _ n +\\dfrac{1}{2} c _ n} \\quad ... [2] \\\\\r\nc _ {n+1} & = \\underline{\\dfrac{1}{2} b _ n +\\dfrac{1}{2} c _ n} \\quad ... [3]\n\\end{align}\\]\r\n

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

    \u8d64\u7389\u306f A , B , C \u306e\u8ab0\u304b\u304c\u6301\u3063\u3066\u3044\u308b\u306e\u3067\r\n\\[\r\na _ n + b _ n + c _ n = 1 \\quad ... [4]\n\\]\r\n[2] \u306b [4] \u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nb _ {n+1} & = \\dfrac{1}{2} -\\dfrac{1}{2} b _ n \\\\\r\n\\text{\u2234} \\quad b _ {n+1} -\\dfrac{1}{3} & = -\\dfrac{1}{2} \\left( b _ n -\\dfrac{1}{3} \\right)\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6570\u5217 \\(\\left\\{ b _ n -\\dfrac{1}{3} \\right\\}\\) \u306f, \u521d\u9805 \\(b _ 1 -\\dfrac{1}{3} =\\dfrac{1}{6}\\) , \u516c\u6bd4 \\(-\\dfrac{1}{2}\\) \u306e\u7b49\u6bd4\u6570\u5217\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nb _ n -\\dfrac{1}{3} & = \\dfrac{1}{6} \\left( -\\dfrac{1}{2} \\right)^{n-1} \\\\\r\n\\text{\u2234} \\quad b _ n & = \\underline{\\dfrac{1}{3} -\\dfrac{1}{3} \\left( -\\dfrac{1}{2} \\right)^n}\n\\end{align}\\]\r\n\u3053\u308c\u3092 [1] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\na _ {n+1} & = \\dfrac{1}{2} a _ n +\\dfrac{1}{6} -\\dfrac{1}{6} \\left( -\\dfrac{1}{2} \\right)^n \\\\\r\n\\text{\u2234} \\quad a _ {n+1} -\\dfrac{1}{3} & -\\dfrac{1}{6} \\left( -\\dfrac{1}{2} \\right)^{n+1} = \\dfrac{1}{2} \\left\\{ a _ n -\\dfrac{1}{3} -\\dfrac{1}{6} \\left( -\\dfrac{1}{2} \\right)^n \\right\\}\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6570\u5217 \\(\\left\\{ a _ n -\\dfrac{1}{3} -\\dfrac{1}{6} \\left( -\\dfrac{1}{2} \\right)^n \\right\\}\\) \u306f, \u521d\u9805 \\(a _ 1 -\\dfrac{1}{3} +\\dfrac{1}{6} \\cdot \\dfrac{1}{2} =\\dfrac{1}{4}\\) , \u516c\u6bd4 \\(\\dfrac{1}{2}\\) \u306e\u7b49\u6bd4\u6570\u5217\u306a\u306e\u3067\r\n\\[\\begin{align}\r\na _ n -\\dfrac{1}{3} & -\\dfrac{1}{6} \\left( -\\dfrac{1}{2} \\right)^n = \\left( \\dfrac{1}{2} \\right)^{n+1} \\\\\r\n\\text{\u2234} \\quad a _ n & = \\underline{\\dfrac{1}{3} -\\dfrac{1}{3} \\left( -\\dfrac{1}{2} \\right)^{n+1} +\\left( \\dfrac{1}{2} \\right)^{n+1}}\n\\end{align}\\]\r\n\u3053\u308c\u3089\u3068 [4] \u3088\u308a\r\n\\[\\begin{align}\r\nc _ n & = 1-a _ n -b _ n \\\\\r\n& = 1 -\\dfrac{1}{3} +\\dfrac{1}{3} \\left( -\\dfrac{1}{2} \\right)^n -\\dfrac{1}{3} +\\dfrac{1}{3} \\left( -\\dfrac{1}{2} \\right)^{n+1} -\\left( \\dfrac{1}{2} \\right)^{n+1} \\\\\r\n& =\\underline{\\dfrac{1}{3} -\\dfrac{1}{3} \\left( -\\dfrac{1}{2} \\right)^{n+1} -\\left( \\dfrac{1}{2} \\right)^{n+1}}\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u306f\u3058\u3081\u306b, A \u304c\u8d64\u7389\u3092 \\(1\\) \u500b, B \u304c\u767d\u7389\u3092 \\(1\\) \u500b, C \u304c\u9752\u7389\u3092 \\(1\\) \u500b\u6301\u3063\u3066\u3044\u308b. \u8868\u88cf\u306e\u51fa\u308b\u78ba\u7387\u304c\u305d\u308c\u305e\u308c \\(\\dfrac{1}{2}\\) \u306e\u786c\u8ca8\u3092\u6295\u3052, \u8868\u304c\u51fa\u308c\u3070 A \u3068 B \u306e … \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":[34],"tags":[143,14],"class_list":["post-452","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2010","tag-nagoya_r","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/452","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=452"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/452\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=452"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=452"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=452"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}