{"id":122,"date":"2011-11-28T00:21:14","date_gmt":"2011-11-27T15:21:14","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=122"},"modified":"2021-09-09T21:19:51","modified_gmt":"2021-09-09T12:19:51","slug":"osr201005","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr201005\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2010\uff1a\u7b2c5\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(n\\) \u3092 \\(0\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b. \u7acb\u65b9\u4f53 ABCD-EFGH \u306e\u9802\u70b9\u3092, \u4ee5\u4e0b\u306e\u3088\u3046\u306b\u79fb\u52d5\u3059\u308b \\(2\\) \u3064\u306e\u52d5\u70b9 P , Q \u3092\u8003\u3048\u308b.\r\n\u6642\u523b \\(0\\) \u306b\u306f P \u306f\u9802\u70b9 A \u306b\u4f4d\u7f6e\u3057, Q \u306f\u9802\u70b9 C \u306b\u4f4d\u7f6e\u3057\u3066\u3044\u308b. \u6642\u523b \\(n\\) \u306b\u304a\u3044\u3066, P \u3068 Q \u304c\u7570\u306a\u308b\u9802\u70b9\u306b\u4f4d\u7f6e\u3057\u3066\u3044\u308c\u3070, \u6642\u523b \\(n+1\\) \u306b\u306f, P \u306f\u6642\u523b \\(n\\) \u306b\u4f4d\u7f6e\u3057\u3066\u3044\u305f\u9802\u70b9\u304b\u3089, \u305d\u308c\u306b\u96a3\u63a5\u3059\u308b \\(3\\) \u9802\u70b9\u306e\u3044\u305a\u308c\u304b\u306b\u7b49\u3057\u3044\u78ba\u7387\u3067\u79fb\u308a, Q \u3082\u6642\u523b \\(n\\) \u306b\u4f4d\u7f6e\u3057\u3066\u3044\u305f\u9802\u70b9\u304b\u3089, \u305d\u308c\u306b\u96a3\u63a5\u3059\u308b \\(3\\) \u9802\u70b9\u306e\u3044\u305a\u308c\u304b\u306b\u7b49\u3057\u3044\u78ba\u7387\u3067\u79fb\u308b.\r\n\u4e00\u65b9, \u6642\u523b \\(n\\) \u306b\u304a\u3044\u3066, P \u3068 Q \u304c\u540c\u3058\u9802\u70b9\u306b\u4f4d\u7f6e\u3057\u3066\u3044\u308c\u3070, \u6642\u523b \\(n+1\\) \u306b\u306f P \u3082 Q \u3082\u6642\u523b \\(n\\) \u306e\u4f4d\u7f6e\u304b\u3089\u306f\u79fb\u52d5\u3057\u306a\u3044.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\u6642\u523b \\(1\\) \u306b\u304a\u3044\u3066, P \u3068 Q \u304c\u7570\u306a\u308b\u9802\u70b9\u306b\u4f4d\u7f6e\u3059\u308b\u3068\u304d, P \u3068 Q \u306f\u3069\u306e\u9802\u70b9\u306b\u3042\u308b\u304b, \u53ef\u80fd\u306a\u7d44\u307f\u5408\u308f\u305b\u3092\u3059\u3079\u3066\u6319\u3052\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\u6642\u523b \\(n\\) \u306b\u304a\u3044\u3066, P \u3068 Q \u304c\u7570\u306a\u308b\u9802\u70b9\u306b\u4f4d\u7f6e\u3059\u308b\u78ba\u7387 \\(r _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\u6642\u523b \\(n\\) \u306b\u304a\u3044\u3066, P \u3068 Q \u304c\u3068\u3082\u306b\u4e0a\u9762 ABCD \u306e\u7570\u306a\u308b\u9802\u70b9\u306b\u4f4d\u7f6e\u3059\u308b\u304b, \u307e\u305f\u306f\u3068\u3082\u306b\u4e0b\u9762 EFGH \u306e\u7570\u306a\u308b\u9802\u70b9\u306b\u4f4d\u7f6e\u3059\u308b\u304b\u306e\u3044\u305a\u308c\u304b\u3067\u3042\u308b\u78ba\u7387\u3092 \\(p _ n\\) \u3068\u3059\u308b.\r\n\u307e\u305f, \u6642\u523b \\(n\\) \u306b\u304a\u3044\u3066, P \u3068 Q \u306e\u3044\u305a\u308c\u304b\u4e00\u65b9\u304c\u4e0a\u9762 ABCD , \u4ed6\u65b9\u304c\u4e0b\u9762 EFGH \u306b\u3042\u308b\u78ba\u7387\u3092 \\(q _ n\\) \u3068\u3059\u308b.\r\n\\(p _ {n+1}\\) \u3092, \\(p _ n\\) \u3068 \\(q _ n\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<li><p><strong>(4)<\/strong>\u3000\\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{q _ n}{p _ n}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\u6642\u523b \\(1\\) \u306b\u304a\u3044\u3066, P \u306f B , D , E \u306e\u3044\u305a\u308c\u304b, Q \u306f B , D , G \u306e\u3044\u305a\u308c\u304b\u306b\u4f4d\u7f6e\u3059\u308b\u306e\u3067, \u6c42\u3081\u308b\u7d44\u5408\u305b\u306f,\r\n\\[\r\n( \\text{P} , \\text{Q} ) = \\underline{( \\text{B} , \\text{D} ), ( \\text{B} , \\text{G} ), ( \\text{D} , \\text{B} ), ( \\text{D} , \\text{G} ), ( \\text{E} , \\text{B} ), ( \\text{E} , \\text{D} ), ( \\text{E} , \\text{G} )}\n\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p><strong>(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nr _ 1 = \\dfrac{7}{9}\n\\]\r\n\u3055\u3089\u306b, P , Q \u304c\u7570\u306a\u308b \\(2\\) \u70b9\u306b\u3042\u308b\u3068\u304d\u306b\u306f\u7acb\u65b9\u4f53\u306e\u3042\u308b\u9762\u306e\u5bfe\u89d2\u7dda\u4e0a\u306b\u4f4d\u7f6e\u3059\u308b\u306e\u3067\r\n\\[\\begin{align}\r\nr _ {n+1} & = \\dfrac{7}{9} r _ n \\\\\r\n\\text{\u2234} \\quad r _ n & = \\underline{\\left( \\dfrac{7}{9} \\right)^n}\n\\end{align}\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n<p>P , Q \u304c\u3068\u3082\u306b\u4e0a\u9762, \u4e0b\u9762\u3044\u305a\u308c\u304b\u306b\u96c6\u307e\u3063\u3066\u3044\u308b\u72b6\u614b\u3092 \\(T\\) , \u5206\u304b\u308c\u3066\u3044\u308b\u72b6\u614b\u3092 \\(S\\) \u3068\u3059\u308b.<\/p>\r\n<ul>\r\n<li><p>\u6642\u523b \\(n \\rightarrow n+1\\) \u306e\u79fb\u52d5\u3067, \u72b6\u614b\u304c \\(S \\rightarrow S\\) \u3068\u306a\u308b\u78ba\u7387\u306f, <strong>(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n\\dfrac{3}{9} = \\dfrac{1}{3}\n\\]<\/li>\r\n<li><p>\u6642\u523b \\(n \\rightarrow n+1\\) \u306e\u79fb\u52d5\u3067, \u72b6\u614b\u304c \\(T \\rightarrow S\\) \u3068\u306a\u308b\u78ba\u7387\u306b\u3064\u3044\u3066\u8003\u3048\u308b.<br \/>\r\n\u6642\u523b \\(n\\) \u306b\u304a\u3044\u3066, \u4eee\u306b P \u304c A , Q \u304c F \u306b\u4f4d\u7f6e\u3059\u308b\u3068\u304d, \u6642\u523b \\(n+1\\) \u306b\u304a\u3044\u3066, P \u306f B , D , E \u306e\u3044\u305a\u308c\u304b, Q \u306f E , G , B \u306e\u3044\u305a\u308c\u304b\u306b\u79fb\u52d5\u3059\u308b.<br \/>\r\n\u3057\u305f\u304c\u3063\u3066, \u72b6\u614b \\(S\\) \u3067\u3042\u308b\u7d44\u5408\u305b\u306f\r\n\\[\r\n( \\text{P} , \\text{Q} ) = ( \\text{D} , \\text{B} ), ( \\text{E} , \\text{G} )\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\dfrac{2}{9}\n\\]<\/li>\r\n<\/ul>\r\n<p>\u4ee5\u4e0a\u3088\u308a\r\n\\[\r\np _ {n+1} = \\underline{\\dfrac{1}{3} p _ n + \\dfrac{2}{9} q _ n}\n\\]\r\n<p><strong>(4)<\/strong><\/p>\r\n<p>\\[\r\np _ n + q _ n = r _ n =\\left( \\dfrac{7}{9} \\right)^n\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\np _ {n+1} & = \\dfrac{1}{3} p _ n + \\dfrac{2}{9} \\left\\{ \\left( \\dfrac{7}{9} \\right)^n -p _ n \\right\\} \\\\\r\n& = \\dfrac{1}{9} p _ n +\\dfrac{2}{9} \\left( \\dfrac{7}{9} \\right)^n \\\\\r\n\\text{\u2234} \\quad & 9^{n+1}p _ {n+1} = 9^n p _ n +2 \\cdot 7^n\n\\end{align}\\]\r\n\\(p _ 0 = 1\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n9^n p _ n & = 9^0 p _ 0 + \\textstyle\\sum\\limits _ {k=0}^{n-1} 2 \\cdot 7^k \\\\\r\n& = 1 +2 \\cdot \\dfrac{7^n -1}{7-1} = \\dfrac{7^n +2}{3} \\\\\r\n\\text{\u2234} \\quad p _ n & = \\dfrac{7^n +2}{3 \\cdot 9^n} = \\dfrac{1}{3} \\left( \\dfrac{7}{9} \\right)^n +\\dfrac{2}{3} \\left( \\dfrac{1}{9} \\right)^n\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nq _ n = \\left( \\dfrac{7}{9} \\right)^n -p _ n = \\dfrac{2}{3} \\left( \\dfrac{7}{9} \\right)^n -\\dfrac{2}{3} \\left( \\dfrac{1}{9} \\right)^n\n\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\n\\dfrac{q _ n}{p _ n} & = \\dfrac{2 \\cdot 7^n -2}{7^n +2} = \\dfrac{2 -2\\left( \\dfrac{1}{7} \\right)^n}{1 +2\\left( \\dfrac{1}{7} \\right)^n} \\\\\r\n& \\rightarrow \\dfrac{2 -2 \\cdot 0}{1 -2 \\cdot 0} = \\underline{2} \\quad ( n \\rightarrow \\infty \\text{\u306e\u3068\u304d} )\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u3092 \\(0\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b. \u7acb\u65b9\u4f53 ABCD-EFGH \u306e\u9802\u70b9\u3092, \u4ee5\u4e0b\u306e\u3088\u3046\u306b\u79fb\u52d5\u3059\u308b \\(2\\) \u3064\u306e\u52d5\u70b9 P , Q \u3092\u8003\u3048\u308b. \u6642\u523b \\(0\\) \u306b\u306f P \u306f\u9802\u70b9 A \u306b\u4f4d\u7f6e\u3057, Q \u306f\u9802\u70b9  &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/osr201005\/\">\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":[31],"tags":[142,14],"class_list":["post-122","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2010","tag-osaka_r","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/122","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=122"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/122\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=122"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=122"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=122"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}