{"id":848,"date":"2013-08-03T13:31:05","date_gmt":"2013-08-03T04:31:05","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=848"},"modified":"2021-09-15T09:28:49","modified_gmt":"2021-09-15T00:28:49","slug":"ngr201301","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201301\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2013\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(3\\) \u4eba\u3067\u30b8\u30e3\u30f3\u30b1\u30f3\u3092\u3059\u308b. \u5404\u4eba\u306f\u30b0\u30fc, \u30c1\u30e7\u30ad, \u30d1\u30fc\u3092\u305d\u308c\u305e\u308c \\(\\dfrac{1}{3}\\) \u306e\u78ba\u7387\u3067\u51fa\u3059\u3082\u306e\u3068\u3059\u308b.\r\n\u8ca0\u3051\u305f\u4eba\u306f\u8131\u843d\u3057, \u6b8b\u3063\u305f\u4eba\u3067\u6b21\u56de\u306e\u30b8\u30e3\u30f3\u30b1\u30f3\u3092\u884c\u3044\uff08\u30a2\u30a4\u30b3\u306e\u3068\u304d\u306f\u8ab0\u3082\u8131\u843d\u3057\u306a\u3044\uff09, \u52dd\u3061\u6b8b\u308a\u304c \\(1\\) \u4eba\u306b\u306a\u308b\u307e\u3067\u30b8\u30e3\u30f3\u30b1\u30f3\u3092\u7d9a\u3051\u308b. \u3053\u306e\u3068\u304d\u5404\u56de\u306e\u8a66\u884c\u306f\u72ec\u7acb\u3068\u3059\u308b. \\(3\\) \u4eba\u3067\u30b8\u30e3\u30f3\u30b1\u30f3\u3092\u59cb\u3081, \u30b8\u30e3\u30f3\u30b1\u30f3\u304c \\(n\\) \u56de\u76ee\u307e\u3067\u7d9a\u3044\u3066 \\(n\\) \u56de\u76ee\u7d42\u4e86\u6642\u306b \\(2\\) \u4eba\u304c\u6b8b\u3063\u3066\u3044\u308b\u78ba\u7387\u3092 \\(p_n\\) , \\(3\\) \u4eba\u304c\u6b8b\u3063\u3066\u3044\u308b\u78ba\u7387\u3092 \\(q_n\\) \u3068\u304a\u304f.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(p_1 , q_1\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(p_n , q_n\\) \u304c\u307f\u305f\u3059\u6f38\u5316\u5f0f\u3092\u5c0e\u304d, \\(p_n , q_n\\) \u306e\u4e00\u822c\u9805\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u3061\u3087\u3046\u3069 \\(n\\) \u56de\u76ee\u3067 \\(1\\) \u4eba\u306e\u52dd\u3061\u6b8b\u308a\u304c\u6c7a\u307e\u308b\u78ba\u7387\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

    \\(3\\) \u4eba\u3067\u30b8\u30e3\u30f3\u30b1\u30f3\u3092\u3059\u308b\u3068\u304d, \\(n\\) \u4eba\uff08 \\(n = 1 , 2\\) \uff09\u304c\u52dd\u3064\u78ba\u7387\u3092 \\(t(n)\\) , \u3042\u3044\u3053\u306b\u306a\u308b\u78ba\u7387\u3092 \\(t(0)\\) \u3068\u304a\u304f.
    \r\n\\(3\\) \u4eba\u306e\u624b\u306e\u51fa\u3057\u65b9\u306f, \\(3^3 = 27\\) \u901a\u308a\u3042\u308b.
    \r\n\\(n = 1 , 2\\) \u306e\u3068\u304d\u306b\u3064\u3044\u3066, \u52dd\u3064\u3068\u304d\u306e\u624b\u306f\u30b0\u30fc, \u30c1\u30e7\u30ad, \u30d1\u30fc\u306e \\(3\\) \u901a\u308a, \u52dd\u3064\u4eba\u306e\u9078\u3073\u65b9\u304c \\({} _ {3}\\text{C} {} _ {n}\\) \u901a\u308a\u3042\u308b\u306e\u3067\r\n\\[ \\begin{align}\r\nt(1) & = \\dfrac{3 \\cdot {} _ {3}\\text{C} {} _ {1}}{3^3} = \\dfrac{1}{3} \\ , \\\\\r\nt(2) & = \\dfrac{3 \\cdot {} _ {3}\\text{C} {} _ {2}}{3^3} = \\dfrac{1}{3} \\ .\r\n\\end{align} \\]\r\n\u3042\u3044\u3053\u306b\u306a\u308b\u306e\u306f, \u3053\u308c\u3089\u52dd\u6557\u304c\u3064\u304f\u3068\u304d\u306e\u4f59\u4e8b\u8c61\u306a\u306e\u3067\r\n\\[\r\nt(0) = 1 -t(1) -t(2) = \\dfrac{1}{3} \\ .\r\n\\]\r\n\u7d9a\u3044\u3066, \\(2\\) \u4eba\u3067\u30b8\u30e3\u30f3\u30b1\u30f3\u3092\u3059\u308b\u3068\u304d, \u52dd\u6557\u304c\u3064\u304f\uff08 \\(1\\) \u4eba\u304c\u52dd\u3064\uff09\u78ba\u7387\u3092 \\(d(1)\\) , \u3042\u3044\u3053\u306b\u306a\u308b\u78ba\u7387\u3092 \\(d(0)\\) \u3068\u304a\u304f.
    \r\n\u540c\u69d8\u306b\u8003\u3048\u308c\u3070\r\n\\[ \\begin{align}\r\nd(1) & = \\dfrac{3 \\cdot {} _ {2}\\text{C} {} _ {1}}{3^2} = \\dfrac{2}{3} \\ , \\\\\r\nd(0) & = 1 -d(1) = \\dfrac{1}{3} \\ .\r\n\\end{align} \\]\r\n

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

    \\[ \\begin{align}\r\np_1 = t(2) = \\underline{\\dfrac{1}{3}} \\ , \\\\\r\nq_1 = t(0) = \\underline{\\dfrac{1}{3}} \\ .\r\n\\end{align} \\]\r\n

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

    \\(n\\) \u56de\u76ee\u7d42\u4e86\u6642\u306b \\(3\\) \u4eba\u304c\u6b8b\u3063\u3066\u3044\u308b\u4e8b\u8c61\u3092 P , \\(2\\) \u4eba\u304c\u6b8b\u3063\u3066\u3044\u308b\u4e8b\u8c61\u3092 Q , \\(1\\) \u4eba\u304c\u6b8b\u3063\u3066\u3044\u308b\u4e8b\u8c61\u3092 R \u3068\u304a\u304f\u3068, \u30b8\u30e3\u30f3\u30b1\u30f3\u306b\u3088\u308b\u72b6\u614b\u306e\u9077\u79fb\u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n\"nagoya_r_2013_01_01\"\r\n

    \u3057\u305f\u304c\u3063\u3066, \\(p_n , q_n\\) \u304c\u307f\u305f\u3059\u6f38\u5316\u5f0f\u306f\r\n\\[ \\begin{align}\r\n& \\left\\{\\begin{array}{l} p_{n+1} = d(0) p_n +t(2) q_{n} \\\\ q_{n+1} = t(0) q_n\\end{array}\\right. \\\\\r\n\\text{\u2234} \\quad & \\underline{\\left\\{\\begin{array}{ll} p_{n+1} = \\dfrac{1}{3} p_n +\\dfrac{1}{3} q_{n} & \\quad ... [1] \\\\ q_{n+1} = \\dfrac{1}{3} q_n & \\quad ... [2] \\end{array}\\right.} \\ .\r\n\\end{align} \\]\r\n[2] \u3068 (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \u6570\u5217 \\(\\{ q_n \\}\\) \u306f, \u521d\u9805 \\(q_1 = \\dfrac{1}{3}\\) , \u516c\u6bd4 \\(\\dfrac{1}{3}\\) \u306e\u7b49\u6bd4\u6570\u5217\u306a\u306e\u3067\r\n\\[\r\nq_n = \\underline{\\dfrac{1}{3^n}} \\ .\r\n\\]\r\n\u3053\u308c\u3092 [1] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[ \\begin{align}\r\np_{n+1} = \\dfrac{1}{3} p_n & +\\dfrac{1}{3^{n+1}} \\\\\r\n\\text{\u2234} \\quad 3^{n+1} p_{n+1} & = 3^n p_n +1 \\ .\r\n\\end{align} \\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6570\u5217 \\(\\{ 3^n p_n \\}\\) \u306f, \u521d\u9805 \\(3 p_1 = 1\\) , \u516c\u5dee \\(1\\) \u306e\u7b49\u5dee\u6570\u5217\u306a\u306e\u3067\r\n\\[ \\begin{align}\r\n3^n p_n & = n \\\\\r\n\\text{\u2234} \\quad p_n & = \\underline{\\dfrac{n}{3^n}} \\ .\r\n\\end{align} \\]\r\n

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

    \\(1\\) \u56de\u76ee\u3067 \\(1\\) \u4eba\u304c\u52dd\u3061\u6b8b\u308b\u306e\u306f\r\n\\[\r\nt(1) = \\dfrac{1}{3} \\ .\r\n\\]\r\n\\(n\\) \u56de\u76ee\uff08 \\(n \\geqq 2\\) \uff09\u3067 \\(1\\) \u4eba\u304c\u52dd\u3061\u6b8b\u308b\u306e\u306f<\/p>\r\n