{"id":742,"date":"2013-04-25T01:35:01","date_gmt":"2013-04-24T16:35:01","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=742"},"modified":"2021-03-10T16:17:31","modified_gmt":"2021-03-10T07:17:31","slug":"tkr201303","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201303\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2013\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

A , B \u306e \\(2\\) \u4eba\u304c\u3044\u308b. \u6295\u3052\u305f\u3068\u304d\u8868\u88cf\u306e\u51fa\u308b\u78ba\u7387\u304c\u305d\u308c\u305e\u308c \\(\\dfrac{1}{2}\\) \u306e\u30b3\u30a4\u30f3\u304c \\(1\\) \u679a\u3042\u308a,\r\n\u6700\u521d\u306f A \u304c\u305d\u306e\u30b3\u30a4\u30f3\u3092\u6301\u3063\u3066\u3044\u308b. \u6b21\u306e\u64cd\u4f5c\u3092\u7e70\u308a\u8fd4\u3059.<\/p>\r\n

    \r\n

  1. (i)<\/strong>\u3000A \u304c\u30b3\u30a4\u30f3\u3092\u6301\u3063\u3066\u3044\u308b\u3068\u304d\u306f, \u30b3\u30a4\u30f3\u3092\u6295\u3052, \u8868\u304c\u51fa\u308c\u3070 A \u306b \\(1\\) \u70b9\u3092\u4e0e\u3048, \u30b3\u30a4\u30f3\u306f A \u304c\u305d\u306e\u307e\u307e\u6301\u3064. \u88cf\u304c\u51fa\u308c\u3070, \u4e21\u8005\u306b\u70b9\u3092\u4e0e\u3048\u305a, A \u306f\u30b3\u30a4\u30f3\u3092 B \u306b\u6e21\u3059.<\/p><\/li>\r\n

  2. (ii)<\/strong>\u3000B \u304c\u30b3\u30a4\u30f3\u3092\u6301\u3063\u3066\u3044\u308b\u3068\u304d\u306f, \u30b3\u30a4\u30f3\u3092\u6295\u3052, \u8868\u304c\u51fa\u308c\u3070 B \u306b \\(1\\) \u70b9\u3092\u4e0e\u3048, \u30b3\u30a4\u30f3\u306f B \u304c\u305d\u306e\u307e\u307e\u6301\u3064. \u88cf\u304c\u51fa\u308c\u3070, \u4e21\u8005\u306b\u70b9\u3092\u4e0e\u3048\u305a, B \u306f\u30b3\u30a4\u30f3\u3092 A \u306b\u6e21\u3059.<\/p><\/li>\r\n<\/ol>\r\n

    \u305d\u3057\u3066A , B \u306e\u3044\u305a\u308c\u304b\u304c \\(2\\) \u70b9\u3092\u7372\u5f97\u3057\u305f\u6642\u70b9\u3067, \\(2\\) \u70b9\u3092\u7372\u5f97\u3057\u305f\u65b9\u306e\u52dd\u5229\u3068\u3059\u308b. \u305f\u3068\u3048\u3070, \u30b3\u30a4\u30f3\u304c\u8868, \u88cf, \u8868, \u8868\u3068\u51fa\u305f\u5834\u5408, \u3053\u306e\u6642\u70b9\u3067 A \u306f \\(1\\) \u70b9, B \u306f \\(2\\) \u70b9\u3092\u7372\u5f97\u3057\u3066\u3044\u308b\u306e\u3067, B \u306e\u52dd\u5229\u3068\u306a\u308b.<\/p>\r\n

      \r\n

    1. (1)<\/strong>\u3000A , B \u3042\u308f\u305b\u3066\u3061\u3087\u3046\u3069 \\(n\\) \u56de\u30b3\u30a4\u30f3\u3092\u6295\u3052\u7d42\u3048\u305f\u3068\u304d\u306b, A \u306e\u52dd\u5229\u3068\u306a\u308b\u78ba\u7387 \\(p _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

    2. (2)<\/strong>\u3000\\(\\sum\\limits _ {n=1}^{\\infty} p(n)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

      \r\n\r\n

      \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

        \r\n

      1. 1*<\/strong>\u3000\\(n=1\\) \u306e\u3068\u304d
        \r\n\u52dd\u6557\u304c\u3064\u304f\u3053\u3068\u306f\u306a\u3044\u306e\u3067\r\n\\[\r\np _ 1 = 0\r\n\\]<\/li>\r\n

      2. 2*<\/strong>\u3000\\(n=2\\) \u306e\u3068\u304d
        \r\n\u300c\u8868\u8868\u300d\u3068\u51fa\u308c\u3070, A \u304c\u52dd\u3064\u306e\u3067\r\n\\[\r\np _ 2 = \\dfrac{1}{2} \\cdot \\dfrac{1}{2} = \\dfrac{1}{4}\r\n\\]<\/li>\r\n

      3. 3*<\/strong>\u3000\\(n=3\\) \u306e\u3068\u304d
        \r\nA \u304c\u52dd\u3064\u3053\u3068\u306f\u306a\u3044\u306e\u3067\r\n\\[\r\np _ 3 = 0\r\n\\]<\/li>\r\n

      4. 4*<\/strong>\u3000\\(n\\) \u304c \\(4\\) \u4ee5\u4e0a\u306e\u5076\u6570\u306e\u3068\u304d
        \r\n\\(1\\) \u56de\u76ee\u304b\u3089 \\(n-1\\) \u56de\u76ee\u307e\u3067\u306b, \u300c\u8868\u300d\u304c \\(1\\) \u56de, \u300c\u88cf\u8868\u300d\u306e\u7d44\u304c \\(\\dfrac{n-2}{2}\\) \u56de\u51fa\u3066, \\(n\\) \u56de\u76ee\u306b\u300c\u8868\u300d\u304c\u51fa\u308c\u3070, A \u304c\u52dd\u3064\u306e\u3067\r\n\\[\r\np _ n = \\dfrac{n}{2} \\cdot \\left( \\dfrac{1}{2} \\right)^n = \\dfrac{n}{2^{n+1}}\r\n\\]\r\n\u3053\u308c\u306f, \\(n=2\\) \u306e\u3068\u304d\u3082\u6e80\u305f\u3057\u3066\u3044\u308b.<\/p><\/li>\r\n

      5. 5*<\/strong>\u3000\\(n\\) \u304c \\(4\\) \u4ee5\u4e0a\u306e\u5947\u6570\u306e\u3068\u304d
        \r\n\\(1\\) \u56de\u76ee\u304b\u3089 \\(n-1\\) \u56de\u76ee\u307e\u3067\u306b, \u300c\u8868\u300d\u304c \\(1\\) \u56de, \u300c\u88cf\u88cf\u8868\u300d\u306e\u7d44\u304c \\(1\\) \u56de,\r\n\u300c\u88cf\u8868\u300d\u306e\u7d44\u304c \\(\\dfrac{n-5}{2}\\) \u56de\u51fa\u3066, \\(n\\) \u56de\u76ee\u306b\u300c\u8868\u300d\u304c\u51fa\u308c\u3070, A \u304c\u52dd\u3064\u306e\u3067\r\n\\[\\begin{align}\r\np _ n & = \\dfrac{n-1}{2} \\cdot \\dfrac{n-3}{2} \\cdot \\left( \\dfrac{1}{2} \\right)^n \\\\\r\n& = \\dfrac{(n-1)(n-3)}{2^{n+2}}\r\n\\end{align}\\]\r\n\u3053\u308c\u306f, \\(n=1, 3\\) \u306e\u3068\u304d\u3082\u6e80\u305f\u3057\u3066\u3044\u308b.<\/p><\/li>\r\n<\/ol>\r\n

        \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u78ba\u7387 \\(p _ n\\) \u306f\r\n\\[\r\np _ n = \\underline{\\left\\{\\begin{array}{ll} \\dfrac{n}{2^{n+1}} & \\left( n \\text{\u304c\u5076\u6570\u306e\u3068\u304d} \\right) \\\\ \\dfrac{(n-1)(n-3)}{2^{n+2}} & \\left( n \\text{\u304c\u5947\u6570\u306e\u3068\u304d} \\right) \\end{array}\\right.}\r\n\\]\r\n

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

        \\(S _ n = \\sum\\limits _ {k=1}^{n} p(k)\\) \u3068\u304a\u304f.<\/p>\r\n

          \r\n

        1. 1*<\/strong>\u3000\\(n\\) \u304c\u5076\u6570\u306e\u3068\u304d
          \r\n\\(n = 2m\\) \uff08 \\(m\\) \u306f\u81ea\u7136\u6570\uff09\u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nS _ {2m} & = \\textstyle\\sum\\limits _ {k=1}^{m} \\left\\{ p(2k-1) +p(2k) \\right\\} \\\\\r\n& = \\textstyle\\sum\\limits _ {k=1}^{m} \\left\\{ \\dfrac{(2k-2)(2k-4)}{2^{2k+1}} +\\dfrac{2k}{2^{2k+1}} \\right\\} \\\\\r\n& = \\textstyle\\sum\\limits _ {k=1}^{m} \\dfrac{2k^2-5k+4}{4^{k}} \\quad ...[1]\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(k\\) \u306b\u3064\u3044\u3066\r\n\\[\r\n\\dfrac{f(k)}{4^{k-1}} -\\dfrac{f(k+1)}{4^{k}} = \\dfrac{2k^2-5k+4}{4^{k+1}} \\quad ...[2]\r\n\\]\r\n\u3092\u6e80\u305f\u3059 \\(x\\) \u306e \\(2\\) \u6b21\u5f0f \\(f(x) = ax^2+bx+c\\) \u3092\u6c42\u3081\u308b.
          \r\n[2] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\n4 ( ak^2+bk+c ) - \\left\\{ ak^2 +(2a+b) k +a +b +c \\right\\} & = 2k^2-5k+4 \\\\\r\n3a k^2 +(3b-2a)k +(3c-a-b) & = 2k^2-5k+4\r\n\\end{align}\\]\r\n\u3053\u308c\u304c \\(k\\) \u306e\u6052\u7b49\u5f0f\u306a\u306e\u3067, \u4fc2\u6570\u3092\u6bd4\u8f03\u3057\u3066\r\n\\[\\begin{align}\r\n3a = 2 , \\quad 3b-2a & = -5 \\quad 3c-a-b = 4 \\\\\r\n\\text{\u2234} \\quad a = \\dfrac{2}{3} , \\quad b & = -\\dfrac{11}{9} , \\quad c = \\dfrac{31}{27}\r\n\\end{align}\\]\r\n[1] \u306b\u5bfe\u3057\u3066, \u3053\u306e\u3053\u3068\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nS _ {2m} & = \\textstyle\\sum\\limits _ {k=1}^{m} \\left\\{ \\dfrac{f(k)}{4^{k-1}} -\\dfrac{f(k+1)}{4^{k}} \\right\\} \\\\\r\n& = f(1) -\\dfrac{f(m+1)}{4^{m}}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067\r\n\\[\r\nf(1) = \\dfrac{2}{3} -\\dfrac{11}{9} +\\dfrac{31}{27} = \\dfrac{16}{27}\r\n\\]\r\n\u307e\u305f, \\(f(m+1)\\) \u306f \\(m\\) \u306e \\(2\\) \u6b21\u5f0f\u306a\u306e\u3067\r\n\\[\r\n\\displaystyle\\lim _ {m \\rightarrow \\infty} \\dfrac{f(m+1)}{4^{m}} = 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066,\r\n\\[\r\n\\displaystyle\\lim _ {m \\rightarrow \\infty} S _ {2m} = \\dfrac{16}{27} \\quad ...[3]\r\n\\]<\/li>\r\n

        2. 2*<\/strong>\u3000\\(n\\) \u304c\u5947\u6570\u306e\u3068\u304d
          \r\n\\(n = 2m-1\\) \uff08 \\(m\\) \u306f\u81ea\u7136\u6570\uff09\u3068\u304a\u304f\u3068\r\n\\[\r\nS _ {2m-1} = S _ {2(m-1)} +p _ {2m-1}\r\n\\]\r\n\u3053\u3053\u3067, [3] \u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {m \\rightarrow \\infty} S _ {2(m-1)} = \\dfrac{16}{27}\r\n\\]\r\n\u307e\u305f,\r\n\\[\r\n\\displaystyle\\lim _ {m \\rightarrow \\infty} p _ {2m-1} = \\displaystyle\\lim _ {m \\rightarrow \\infty} \\dfrac{2m-1}{2^{2m}} = 0\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\displaystyle\\lim _ {m \\rightarrow \\infty} S _ {2m-1} = \\dfrac{16}{27} \\quad ...[4]\r\n\\]\r\n\u3088\u3063\u3066, [3] [4] \u304b\u3089 \\(\\displaystyle\\lim _ {m \\rightarrow \\infty} S _ {n}\\) \u306f \\(n\\) \u306e\u5947\u5076\u306b\u95a2\u308f\u3089\u305a\u540c\u3058\u5024\u306b\u53ce\u675f\u3057\r\n\\[\r\n\\textstyle\\sum\\limits _ {n=1}^{\\infty} p(n) = \\displaystyle\\lim _ {m \\rightarrow \\infty} S _ {n} = \\underline{\\dfrac{16}{27}}\r\n\\]<\/li>\r\n<\/ol>\r\n","protected":false},"excerpt":{"rendered":"A , B \u306e \\(2\\) \u4eba\u304c\u3044\u308b. \u6295\u3052\u305f\u3068\u304d\u8868\u88cf\u306e\u51fa\u308b\u78ba\u7387\u304c\u305d\u308c\u305e\u308c \\(\\dfrac{1}{2}\\) \u306e\u30b3\u30a4\u30f3\u304c \\(1\\) \u679a\u3042\u308a, \u6700\u521d\u306f A \u304c\u305d\u306e\u30b3\u30a4\u30f3\u3092\u6301\u3063\u3066\u3044\u308b. \u6b21\u306e\u64cd\u4f5c\u3092\u7e70\u308a\u8fd4\u3059. (i)\u3000A \u304c … \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":[86],"tags":[139,111],"class_list":["post-742","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2013","tag-tokyo_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/742","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=742"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/742\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=742"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=742"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=742"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}