{"id":1278,"date":"2015-09-12T01:19:01","date_gmt":"2015-09-11T16:19:01","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1278"},"modified":"2021-09-14T20:50:09","modified_gmt":"2021-09-14T11:50:09","slug":"ngr201504","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201504\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2015\uff1a\u7b2c4\u554f"},"content":{"rendered":"<hr \/>\n<p>\u6570\u76f4\u7dda\u4e0a\u306b\u3042\u308b \\(1, 2, 3, 4, 5\\) \u306e \\(5\\) \u3064\u306e\u70b9\u3068 \\(1\\) \u3064\u306e\u77f3\u3092\u8003\u3048\u308b. \u77f3\u304c\u3044\u305a\u308c\u304b\u306e\u70b9\u306b\u3042\u308b\u3068\u304d,<\/p>\r\n<ul>\r\n<li><p>\u77f3\u304c\u70b9 \\(1\\) \u306b\u3042\u308b\u306a\u3089\u3070, \u78ba\u7387 \\(1\\) \u3067\u70b9 \\(2\\) \u306b\u79fb\u52d5\u3059\u308b.<\/p><\/li>\r\n<li><p>\u77f3\u304c\u70b9 \\(k \\ ( k = 2, 3, 4 )\\) \u306b\u3042\u308b\u306a\u3089\u3070, \u78ba\u7387 \\(\\dfrac{1}{2}\\) \u3067\u70b9 \\(k+1\\) \u306b\u79fb\u52d5\u3059\u308b.<\/p><\/li>\r\n<li><p>\u77f3\u304c\u70b9 \\(5\\) \u306b\u3042\u308b\u306a\u3089\u3070, \u78ba\u7387 \\(1\\) \u3067\u70b9 \\(4\\) \u306b\u79fb\u52d5\u3059\u308b.<\/p><\/li>\r\n<\/ul>\r\n<p>\u3068\u3044\u3046\u8a66\u884c\u3092\u884c\u3046. \u77f3\u304c\u70b9 \\(1\\) \u306b\u3042\u308b\u72b6\u614b\u304b\u3089\u59cb\u3081, \u3053\u306e\u8a66\u884c\u3092\u7e70\u308a\u8fd4\u3059. \u307e\u305f, \u77f3\u304c\u79fb\u52d5\u3057\u305f\u5148\u306e\u70b9\u306b\u5370\u3092\u3064\u3051\u3066\u3044\u304f\uff08\u70b9 \\(1\\) \u306b\u306f\u521d\u3081\u304b\u3089\u5370\u304c\u3064\u3044\u3066\u3044\u308b\u3082\u306e\u3068\u3059\u308b\uff09. \u3053\u306e\u3068\u304d, \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\u8a66\u884c\u3092 \\(6\\) \u56de\u7e70\u308a\u8fd4\u3057\u305f\u5f8c\u306b, \u77f3\u304c\u70b9 \\(k \\ ( k = 1, 2, 3, 4, 5 )\\) \u306b\u3042\u308b\u78ba\u7387\u3092\u305d\u308c\u305e\u308c\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\u8a66\u884c\u3092 \\(6\\) \u56de\u7e70\u308a\u8fd4\u3057\u305f\u5f8c\u306b, \\(5\\) \u3064\u306e\u3059\u3079\u3066\u306b\u5370\u304c\u3064\u3044\u3066\u3044\u308b\u78ba\u7387\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\u8a66\u884c\u3092 \\(n\\) \u56de\uff08 \\(n \\geqq 1\\) \uff09\u7e70\u308a\u8fd4\u3057\u305f\u5f8c\u306b, \u3061\u3087\u3046\u3069 \\(3\\) \u3064\u306e\u70b9\u306b\u5370\u304c\u3064\u3044\u3066\u3044\u308b\u78ba\u7387\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>\\(n\\) \u56de\u306e\u8a66\u884c\u5f8c\u306b, \u5404\u70b9\u306b\u77f3\u304c\u3042\u308b\u78ba\u7387\u3092\u8868\u306b\u3059\u308b\u3068, \u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\uff08\u7a7a\u6b04\u306f, \u78ba\u7387 \\(0\\) \uff09.\r\n\\[\r\n\\begin{array}{c||c|c|c|c|c|c} \\text{\u70b9\uff5c\u8a66\u884c} & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline \\hline 1 & & \\dfrac{1}{2} & & \\dfrac{3}{8} & & \\dfrac{5}{16} \\\\ \\hline 2 & 1 & & \\dfrac{3}{4} & & \\dfrac{5}{8} & \\\\ \\hline 3 & & \\dfrac{1}{2} & & \\dfrac{1}{2} & & \\dfrac{1}{2} \\\\ \\hline 4 & & & \\dfrac{1}{4} & & \\dfrac{3}{8} & \\\\ \\hline 5 & & & & \\dfrac{1}{8} & & \\dfrac{3}{16} \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u70b9 \\(1, 2, \\cdots , 5\\) \u306b\u3042\u308b\u78ba\u7387\u306f\u9806\u306b\r\n\\[\r\n\\underline{\\dfrac{5}{16} , \\ 0 , \\ \\dfrac{1}{2} , \\ 0 , \\ \\dfrac{3}{16}} \\ .\r\n\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\u6761\u4ef6\u3092\u6e80\u305f\u3059\u306e\u306f, \u4ee5\u4e0b\u306e \\(2\\) \u3064\u306e\u5834\u5408\u3067\u3042\u308b.<\/p>\r\n<ul>\r\n<li><p>\\(4\\) \u56de\u76ee\u306b\u70b9 \\(5\\) \u306b\u5230\u9054\u3059\u308b.<\/p><\/li>\r\n<li><p>\\(4\\) \u56de\u76ee\u306b\u70b9 \\(3\\) \u306b\u3042\u308a, \u305d\u306e\u5f8c \\(6\\) \u56de\u76ee\u306b\u70b9 \\(5\\) \u306b\u5230\u9054\u3059\u308b.<\/p><\/li>\r\n<\/ul>\r\n<p>\u3088\u3063\u3066, \u6c42\u3081\u308b\u78ba\u7387\u306f\r\n\\[\r\n\\dfrac{1}{8} +\\dfrac{1}{2} \\cdot \\left( \\dfrac{1}{2} \\right)^2 = \\underline{\\dfrac{1}{4}} \\ .\r\n\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n<p>\u6c42\u3081\u308b\u78ba\u7387\u3092 \\(P _ n\\) \u3068\u304a\u304f.<br \/>\r\n\u77f3\u306f, \u5947\u6570\u56de\u5f8c\u306b\u306f\u70b9 \\(2, 4\\) \u306b, \u5076\u6570\u56de\u5f8c\u306b\u306f\u70b9 \\(1, 3, 5\\) \u306b\u3042\u308b\u3053\u3068\u306b\u7740\u76ee\u3057, \u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(n\\) \u304c\u5947\u6570\u306e\u3068\u304d<br \/>\r\n\u300c \\(n\\) \u56de\u5f8c\u306b\u70b9 \\(2\\) \u306b\u3042\u308b\u77f3\u304c, \\(n+2\\) \u56de\u5f8c\u306b\u70b9 \\(2\\) \u306b\u623b\u3063\u3066\u304f\u308b\u300d ... [1] \u78ba\u7387\u306f, \\(\\dfrac{3}{4}\\) .<br \/>\r\n\\(n = 1\\) \u4ee5\u964d, [1] \u3092\u7e70\u308a\u8fd4\u3057\u3066\u3044\u308c\u3070, \u5370\u304c\u3064\u304f\u70b9\u306f \\(1, 2, 3\\) \u306e\u307f\u3067\u3042\u308b.<br \/>\r\n\u305f\u3060\u3057, [1] \u306b\u306f, \u77f3\u304c\u70b9 \\(1, 2\\) \u306e\u307f\u306e\u9593\u3092\u79fb\u52d5\u3059\u308b\u5834\u5408\u304c\u542b\u307e\u308c\u3066\u3044\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308c\u3070\r\n\\[\r\nP _ n = \\left( \\dfrac{3}{4} \\right)^{\\frac{n-1}{2}} -\\left( \\dfrac{1}{2} \\right)^{\\frac{n-1}{2}} \\ .\r\n\\]<\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(n\\) \u304c\u5076\u6570\u306e\u3068\u304d<br \/>\r\n\\(n-1\\) \u56de\u5f8c\u307e\u3067, <strong>1*<\/strong> \u306e\u5834\u5408\u3092\u307f\u305f\u3059\u3088\u3046\u306b, \u70b9\u304c\u79fb\u52d5\u3057\u3066\u3044\u308c\u3070, \\(n\\) \u56de\u76ee\u306e\u8a66\u884c\u306e\u7d50\u679c\u306b\u3088\u3089\u305a, \u6761\u4ef6\u3092\u307f\u305f\u3059\u306e\u3067\r\n\\[\r\nP _ n = P _ {n-1} = \\left( \\dfrac{3}{4} \\right)^{\\frac{n}{2} -1} -\\left( \\dfrac{1}{2} \\right)^{\\frac{n}{2} -1} \\ .\r\n\\]\r\n\u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u78ba\u7387\u306f\r\n\\[\r\nP _ n = \\underline{\\left\\{ \\begin{array}{ll} \\left( \\dfrac{3}{4} \\right)^{\\frac{n-1}{2}} -\\left( \\dfrac{1}{2} \\right)^{\\frac{n-1}{2}} & ( \\ n \\ \\text{\u304c\u5947\u6570\u306e\u3068\u304d} ) \\\\ \\left( \\dfrac{3}{4} \\right)^{\\frac{n}{2} -1} -\\left( \\dfrac{1}{2} \\right)^{\\frac{n}{2} -1} & ( \\ n \\ \\text{\u304c\u5076\u6570\u306e\u3068\u304d} ) \\end{array} \\right.} \\ .\r\n\\]<\/li>\r\n<\/ol>\r\n<h2>\u3010 \u53c2 \u8003 \u3011<\/h2>\r\n<p>\\(n\\) \u56de\u306e\u8a66\u884c\u5f8c\u306e\u70b9 \\(1, 2, \\cdots , 5\\) \u306b\u3042\u308b\u78ba\u7387\u3092 \\(p _ n , q _ n , \\cdots , t _ n\\) \u3068\u304a\u304f\u3068<\/p>\r\n<ul>\r\n<li><p>\\(n\\) \u304c\u5947\u6570\u306e\u3068\u304d<br \/>\r\n\\(q _ n +s _ n = 1\\) \u306a\u306e\u3067, \\(s _ n = 1 -q _ n\\) \u306a\u306e\u3067, \\(n \\geqq 1\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nq _ {n+2} & = \\dfrac{3}{4} q _ n +\\dfrac{1}{4} s _ n \\\\\r\n& = \\dfrac{1}{2} q _ n +\\dfrac{1}{4} , \\\\\r\n\\text{\u2234} \\quad q _ {n+2} -\\dfrac{1}{2} & = \\dfrac{1}{2} \\left( q _ n -\\dfrac{1}{2} \\right) \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u3053\u308c\u3092\u7e70\u8fd4\u3057\u7528\u3044\u308c\u3070\r\n\\[\r\nq _ n -\\dfrac{1}{2} = \\left( q _ 1 -\\dfrac{1}{2} \\right) \\left( \\dfrac{1}{2} \\right)^{\\frac{n-1}{2}} = \\left( \\dfrac{1}{2} \\right)^{\\frac{n+1}{2}} \\ .\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\\begin{align}\r\nq _ n & = \\dfrac{1}{2} +\\left( \\dfrac{1}{2} \\right)^{\\frac{n+1}{2}} , \\\\\r\ns _ n & = 1-q _ n = \\dfrac{1}{2} -\\left( \\dfrac{1}{2} \\right)^{\\frac{n+1}{2}} \\ .\r\n\\end{align}\\]<\/li>\r\n<li><p>\\(n\\) \u304c\u5076\u6570\u306e\u3068\u304d<br \/>\r\n\\(p _ n +r _ n +t _ n = 1\\) \u306b\u7740\u76ee\u3059\u308c\u3070, \\(n \\geqq 2\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nr _ {n+2} & = \\dfrac{1}{2} p _ n +\\dfrac{1}{2} r _ n +\\dfrac{1}{2} t _ n = \\dfrac{1}{2} , \\\\\r\np _ {n+2} & = \\dfrac{1}{2} p _ n +\\dfrac{1}{4} r _ n = \\dfrac{1}{2} p _ n +\\dfrac{1}{8} , \\\\\r\nt _ {n+2} & = \\dfrac{1}{2} t _ n +\\dfrac{1}{4} r _ n = \\dfrac{1}{2} t _ n +\\dfrac{1}{8} \\ .\r\n\\end{align}\\]\r\n\u5909\u5f62\u3059\u308b\u3068\r\n\\[\\begin{align}\r\np _ {n+2} -\\dfrac{1}{4} & = \\dfrac{1}{2} \\left( p _ n -\\dfrac{1}{4} \\right) , \\\\\r\nt _ {n+2} -\\dfrac{1}{4} & = \\dfrac{1}{2} \\left( t _ n -\\dfrac{1}{4} \\right) \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\np _ n -\\dfrac{1}{4} & = \\left( p _ 2 -\\dfrac{1}{4} \\right) \\left( \\dfrac{1}{2} \\right)^{\\frac{n}{2} -1} = \\left( \\dfrac{1}{2} \\right)^{\\frac{n}{2} +1} \\\\\r\n& \\text{\u2234} \\quad p _ n = \\dfrac{1}{4} +\\left( \\dfrac{1}{2} \\right)^{\\frac{n}{2} +1} \\\\\r\nt _ n -\\dfrac{1}{4} & = \\left( r _ 2 -\\dfrac{1}{4} \\right) \\left( \\dfrac{1}{2} \\right)^{\\frac{n}{2} -1} = -\\left( \\dfrac{1}{2} \\right)^{\\frac{n}{2} +1} \\\\\r\n& \\text{\u2234} \\quad t _ n = \\dfrac{1}{4} -\\left( \\dfrac{1}{2} \\right)^{\\frac{n}{2} +1} \\ .\r\n\\end{align}\\]<\/li>\r\n<\/ul>\r\n<p>\u4ee5\u4e0a\u3088\u308a\r\n\\[\r\n\\begin{array}{c||c|c} n & \\text{\u5947\u6570\u306e\u3068\u304d} & \\text{\u5076\u6570\u306e\u3068\u304d} \\\\ \\hline p _ n & \\dfrac{1}{4} +\\left( \\dfrac{1}{2} \\right)^{\\frac{n}{2} +1} & 0 \\\\ \\hline q _ n & 0 & \\dfrac{1}{2} +\\left( \\dfrac{1}{2} \\right)^{\\frac{n+1}{2}} \\\\ \\hline r _ n & \\dfrac{1}{2} & 0 \\\\ \\hline s _ n & 0 & \\dfrac{1}{2} -\\left( \\dfrac{1}{2} \\right)^{\\frac{n+1}{2}} \\\\ \\hline t _ n & \\dfrac{1}{4} -\\left( \\dfrac{1}{2} \\right)^{\\frac{n}{2} +1} & 0 \\end{array}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6570\u76f4\u7dda\u4e0a\u306b\u3042\u308b \\(1, 2, 3, 4, 5\\) \u306e \\(5\\) \u3064\u306e\u70b9\u3068 \\(1\\) \u3064\u306e\u77f3\u3092\u8003\u3048\u308b. \u77f3\u304c\u3044\u305a\u308c\u304b\u306e\u70b9\u306b\u3042\u308b\u3068\u304d, \u77f3\u304c\u70b9 \\(1\\) \u306b\u3042\u308b\u306a\u3089\u3070, \u78ba\u7387 \\(1\\) \u3067\u70b9 \\(2\\) \u306b\u79fb\u52d5\u3059\u308b &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/ngr201504\/\">\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":[129],"tags":[143,137],"class_list":["post-1278","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2015","tag-nagoya_r","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1278","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=1278"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1278\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1278"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1278"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1278"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}