{"id":73,"date":"2011-11-26T22:32:13","date_gmt":"2011-11-26T13:32:13","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=73"},"modified":"2021-11-04T09:31:35","modified_gmt":"2021-11-04T00:31:35","slug":"htb201105","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb201105\/","title":{"rendered":"\u4e00\u6a4b\u59272011\uff1a\u7b2c5\u554f"},"content":{"rendered":"<hr \/>\n<p>A \u3068 B \u306e \\(2\\) \u4eba\u304c, \\(1\\) \u500b\u306e\u30b5\u30a4\u30b3\u30ed\u3092\u6b21\u306e\u624b\u9806\u306b\u3088\u308a\u6295\u3052\u5408\u3046.<\/p>\r\n<ul>\r\n<li><p>\\(1\\) \u56de\u76ee\u306f A \u304c\u6295\u3052\u308b.<\/p><\/li>\r\n<li><p>\\(1, 2, 3\\) \u306e\u76ee\u304c\u51fa\u305f\u3089, \u6b21\u306e\u56de\u306b\u306f\u540c\u3058\u4eba\u304c\u6295\u3052\u308b.<\/p><\/li>\r\n<li><p>\\(4, 5\\) \u306e\u76ee\u304c\u51fa\u305f\u3089, \u6b21\u306e\u56de\u306b\u306f\u5225\u306e\u4eba\u304c\u6295\u3052\u308b.<\/p><\/li>\r\n<li><p>\\(6\\) \u306e\u76ee\u304c\u51fa\u305f\u3089, \u6295\u3052\u305f\u4eba\u3092\u52dd\u3061\u3068\u3057\u305d\u308c\u4ee5\u964d\u306f\u6295\u3052\u306a\u3044.<\/p><\/li>\r\n<\/ul>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(n\\) \u56de\u76ee\u306b A \u304c\u30b5\u30a4\u30b3\u30ed\u3092\u6295\u3052\u308b\u78ba\u7387 \\(a _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\u3061\u3087\u3046\u3069 \\(n\\) \u56de\u76ee\u306e\u30b5\u30a4\u30b3\u30ed\u6295\u3052\u3067 A \u304c\u52dd\u3064\u78ba\u7387 \\(p _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\\(n\\) \u56de\u4ee5\u5185\u306e\u30b5\u30a4\u30b3\u30ed\u6295\u3052\u3067 A \u304c\u52dd\u3064\u78ba\u7387 \\(q _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\\(n\\) \u56de\u76ee\u306b A , B \u304c\u6295\u3052\u308b\u78ba\u7387\u3092\u305d\u308c\u305e\u308c \\(a _ n\\) , \\(b _ n\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\na _ {n+1} & = \\dfrac{1}{2} a _ n +\\dfrac{1}{3} b _ n \\quad ... [1] , \\\\\r\nb _ {n+1} & = \\dfrac{1}{3} a _ n +\\dfrac{1}{2} b _ n \\quad ... [2]\r\n\\end{align}\\]\r\n\\([1]+[2]\\) \u3088\u308a\r\n\\[\r\na _ {n+1}+b _ {n+1} = \\dfrac{5}{6} ( a _ n+b _ n )\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\na _ n+b _ n = \\left( \\dfrac{5}{6} \\right)^{n-1} ( a _ 1+b _ 1 ) = \\left( \\dfrac{5}{6} \\right)^{n-1} \\quad ... [3]\r\n\\]\r\n\\([1]-[2]\\) \u3088\u308a\r\n\\[\r\na _ {n+1}-b _ {n+1} = \\dfrac{1}{6} ( a _ n-b _ n )\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\na _ n-b _ n = \\left( \\dfrac{1}{6} \\right)^{n-1} ( a _ 1-b _ 1 ) = \\left( \\dfrac{1}{6} \\right)^{n-1} \\quad ... [4]\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [3] [4] \u3088\u308a\r\n\\[\r\na _ n = \\underline{\\dfrac{1}{2} \\left\\{ \\left( \\dfrac{5}{6} \\right)^{n-1} +\\left( \\dfrac{1}{6} \\right)^{n-1} \\right\\}}\r\n\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\\(n\\) \u56de\u76ee\u306b A \u304c\u6295\u3052\u3066, \\(6\\) \u304c\u51fa\u308c\u3070\u3088\u3044\u306e\u3067\r\n\\[\r\np _ n =\\dfrac{1}{6} a _ n = \\underline{\\dfrac{1}{12} \\left\\{ \\left( \\dfrac{5}{6} \\right)^{n-1} +\\left( \\dfrac{1}{6} \\right)^{n-1} \\right\\}}\r\n\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n<p><strong>(2)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nq _ n & = \\textstyle\\sum\\limits _ {k=1}^n p _ n = \\dfrac{1}{12} \\textstyle\\sum\\limits _ {k=1}^n \\left\\{ \\left( \\dfrac{5}{6} \\right)^{k-1} +\\left( \\dfrac{1}{6} \\right)^{k-1} \\right\\} \\\\\r\n& = \\dfrac{1}{12} \\cdot \\dfrac{1 -\\left( \\dfrac{5}{6} \\right)^n}{1 -\\dfrac{5}{6}} +\\dfrac{1}{12} \\cdot \\dfrac{1 -\\left( \\dfrac{1}{6} \\right)^n}{1 -\\dfrac{1}{6}} \\\\\r\n& = \\dfrac{1}{2} \\left\\{1 -\\left( \\dfrac{5}{6} \\right)^n \\right\\} +\\dfrac{1}{10} \\left\\{ 1 -\\left( \\dfrac{1}{6} \\right)^n \\right\\} \\\\\r\n& = \\underline{\\dfrac{3}{5} -\\dfrac{1}{2} \\left( \\dfrac{5}{6} \\right)^n +\\dfrac{1}{10} \\left( \\dfrac{1}{6} \\right)^n}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"A \u3068 B \u306e \\(2\\) \u4eba\u304c, \\(1\\) \u500b\u306e\u30b5\u30a4\u30b3\u30ed\u3092\u6b21\u306e\u624b\u9806\u306b\u3088\u308a\u6295\u3052\u5408\u3046. \\(1\\) \u56de\u76ee\u306f A \u304c\u6295\u3052\u308b. \\(1, 2, 3\\) \u306e\u76ee\u304c\u51fa\u305f\u3089, \u6b21\u306e\u56de\u306b\u306f\u540c\u3058\u4eba\u304c\u6295\u3052\u308b. \\(4, 5\\) \u306e\u76ee\u304c\u51fa\u305f &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/htb201105\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[47],"tags":[146,13],"class_list":["post-73","post","type-post","status-publish","format-standard","hentry","category-hitotsubashi_2011","tag-hitotsubashi","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/73","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=73"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/73\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=73"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=73"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=73"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}