{"id":31,"date":"2011-11-25T21:46:17","date_gmt":"2011-11-25T12:46:17","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=31"},"modified":"2021-09-09T20:48:41","modified_gmt":"2021-09-09T11:48:41","slug":"osr201105","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr201105\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2011\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\u6b63\u6570 \\(r\\) \u306b\u5bfe\u3057\u3066, \\(a _ 1 = 0\\) , \\(a _ 2 = r\\) \u3068\u304a\u304d, \u6570\u5217 \\(\\{ a _ n \\}\\) \u3092\u6b21\u306e\u6f38\u5316\u5f0f\u3067\u5b9a\u3081\u308b.\r\n\\[\r\na _ {n+1} = a _ n +r _ n ( a _ n -a _ {n-1} ) \\quad ( n = 2, 3, 4, \\cdots )\n\\]\r\n\u305f\u3060\u3057 \\(a _ n\\) \u3068 \\(a _ {n-1}\\) \u304b\u3089\u6f38\u5316\u5f0f\u3092\u7528\u3044\u3066 \\(a _ {n+1}\\) \u3092\u6c7a\u3081\u308b\u969b\u306b\u306f\u786c\u8ca8\u3092\u6295\u3052, \u8868\u304c\u3067\u305f\u3068\u304d \\(r _ n = \\dfrac{r}{2}\\) , \u88cf\u304c\u3067\u305f\u3068\u304d \\(r _ n = \\dfrac{1}{2r}\\) \u3068\u3059\u308b. \u3053\u3053\u3067\u8868\u304c\u3067\u308b\u78ba\u7387\u3068\u88cf\u304c\u3067\u308b\u78ba\u7387\u306f\u7b49\u3057\u3044\u3068\u3059\u308b. \\(a _ n\\) \u306e\u671f\u5f85\u5024\u3092 \\(p _ n\\) \u3068\u3059\u308b\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(p _ 3\\) \u304a\u3088\u3073 \\(p _ 4\\) \u3092, \\(r\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(n \\geqq 3\\) \u306e\u3068\u304d\u306b \\(p _ n\\) \u3092, \\(n\\) \u3068 \\(r\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u6570\u5217 \\(\\{ p _ n \\}\\) \u304c\u53ce\u675f\u3059\u308b\u3088\u3046\u306a\u6b63\u6570 \\(r\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  4. (4)<\/strong>\u3000\\(r\\) \u304c (3)<\/strong> \u3067\u6c42\u3081\u305f\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \u6975\u9650\u5024 \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} p _ n\\) \u306e\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

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

    \\(b _ n = a _ {n+1} -a _ n \\ ( n= 2, 3, \\cdots )\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nb _ n = r _ n b _ {n-1} , \\ b _ 1 = a _ 2 -a _ 1 = r\n\\]\r\n\u3053\u308c\u3092\u7e70\u8fd4\u3057\u7528\u3044\u308c\u3070, \\(n \\geqq 2\\) \u306e\u3068\u304d\r\n\\[\r\nb _ n = r _ n \\cdot r _ {n-1} \\cdots r _ 2 \\cdot r\n\\]\r\n\u3053\u3053\u3067, \\(r _ 2, \\cdots , r _ {n-1}\\) \u306e\u5024\u306e\u7d44\u5408\u305b\u306f, \\(2^{n-1}\\) \u901a\u308a\u3042\u308a\u540c\u69d8\u306b\u78ba\u304b\u3089\u3057\u3044.
    \r\n\u305d\u308c\u305e\u308c\u306e\u5834\u5408\u306b\u304a\u3051\u308b \\(b _ n\\) \u306e\u5024\u3092 \\(c _ i \\ ( i = 1 , \\cdots , 2^{n-1} )\\) \u3068\u304a\u3051\u3070, \\(b _ n\\) \u306e\u671f\u5f85\u5024\uff08\u3059\u306a\u308f\u3061 \\(p _ {n+1}-p _ n\\) \uff09\u3092 \\(q _ n\\) \u3068\u304a\u3051\u3070, \u4e8c\u9805\u5b9a\u7406\u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\nq _ n & = p _ {n+1}-p _ n = \\textstyle\\sum\\limits _ {i=1}^{2^{n-1}} \\dfrac{c _ i}{2^{n-1}} \\\\\r\n& = \\dfrac{r}{2^{n-1}} \\textstyle\\sum\\limits _ {k=0}^{n-1} {} _ {n-1} \\text{C} {} _ k \\left( \\dfrac{r}{2} \\right)^k \\left( \\dfrac{1}{2r} \\right)^{n-1-k} \\\\\r\n& = \\dfrac{r}{2^{n-1}} \\left( \\dfrac{r}{2} +\\dfrac{1}{2r} \\right)^{n-1} \\\\\r\n& = r \\left( \\dfrac{r}{4}+\\dfrac{1}{4r} \\right)^{n-1} \\quad ... [1]\n\\end{align}\\]\r\n[1] \u3092\u7528\u3044\u308c\u3070,\r\n\\[\\begin{align}\r\np _ 3 & = p _ 2 +q _ 2 = a _ 2 +r \\left( \\dfrac{r}{4}+\\dfrac{1}{4r} \\right) \\\\\r\n& = \\underline{\\dfrac{r^2}{4} +r +\\dfrac{1}{4}} , \\\\\r\np _ 4 & = p _ 3 +q _ 3 = \\dfrac{r^2}{4} +r +\\dfrac{1}{4} +r \\left( \\dfrac{r}{4}+\\dfrac{1}{4r} \\right)^2 \\\\\r\n& = \\dfrac{r^2}{4} +r +\\dfrac{1}{4} +r \\left( \\dfrac{r^2}{16} +\\dfrac{1}{8} +\\dfrac{1}{16r^2} \\right) \\\\\r\n& = \\underline{\\dfrac{r^3}{16} +\\dfrac{r^2}{4} +\\dfrac{9r}{8} +\\dfrac{1}{4} +\\dfrac{1}{16r}}\n\\end{align}\\]\r\n

    (2)<\/strong><\/p>\r\n[1] \u3088\u308a, \\(n \\geqq 3\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\np _ n & = p _ 2 +\\textstyle\\sum\\limits _ {k=2}^{n-1} q _ k \\\\\r\n& = r \\textstyle\\sum\\limits _ {k=1}^{n-1} \\left( \\dfrac{r}{4}+\\dfrac{1}{4r} \\right)^{k-1} \\quad ... [2]\n\\end{align}\\]\r\n\u6570\u5217 \\(\\{ q _ n \\}\\) \u306e\u516c\u6bd4\u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{r}{4} +\\dfrac{1}{4r} & = 1 \\\\\r\nr^2 -4r +1 & = 0 \\\\\r\n\\text{\u2234} \\quad r & = 2 \\pm \\sqrt{3}\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(r \\neq 2 \\pm \\sqrt{3}\\) \u306e\u3068\u304d, [2]\u3088\u308a\r\n\\[\r\np _ n = r \\cdot \\dfrac{1-\\left( \\dfrac{r}{4}+\\dfrac{1}{4r} \\right)^{n-1}}{1-\\left( \\dfrac{r}{4}+\\dfrac{1}{4r} \\right)}\n\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(r = 2 \\pm \\sqrt{3}\\) \u306e\u3068\u304d, [2]\u3088\u308a\r\n\\[\r\np _ n = r(n-1)\n\\]<\/li>\r\n<\/ol>\r\n

      1*<\/strong> 2*<\/strong>\u3088\u308a\r\n\\[\r\np _ n = \\underline{\\left\\{ \\begin{array}{ll} r \\cdot \\dfrac{1-\\left( \\dfrac{r}{4}+\\dfrac{1}{4r} \\right)^{n-1}}{1-\\left( \\dfrac{r}{4}+\\dfrac{1}{4r} \\right)} & ( r \\neq 2 \\pm \\sqrt{3} \\text{\u306e\u3068\u304d} ) \\\\ r(n-1) & ( r = 2 \\pm \\sqrt{3} \\text{\u306e\u3068\u304d} ) \\end{array} \\right.}\n\\]\r\n

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

      \\(p _ n\\) \u304c\u53ce\u675f\u3059\u308b\u6761\u4ef6\u306f, \\(r \\gt 0\\) \u306a\u306e\u3067\r\n\\[\\begin{gather}\r\n\\dfrac{r}{4} +\\dfrac{1}{4r} \\lt 1 \\\\\r\n\\text{\u2234} \\quad \\underline{2-\\sqrt{3} \\lt r \\lt 2+\\sqrt{3}}\n\\end{gather}\\]\r\n

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

      (3)<\/strong> \u306e\u5834\u5408\u306b\u306f,\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} p _ n & = \\dfrac{r}{1-\\left( \\dfrac{r}{4}+\\dfrac{1}{4r} \\right)} \\\\\r\n& = \\dfrac{4}{-\\dfrac{1}{r^2}+\\dfrac{4}{r}-1} \\\\\r\n& = \\dfrac{4}{-\\left( \\dfrac{1}{r}-2 \\right)^2 +3} \\leqq \\dfrac{4}{3}\n\\end{align}\\]\r\n\u7b49\u53f7\u6210\u7acb\u306f \\(\\dfrac{1}{r} =2\\) \u3059\u306a\u308f\u3061 \\(r=\\dfrac{1}{2}\\) \u306e\u3068\u304d.
      \r\n\u3086\u3048\u306b, \u6c42\u3081\u308b\u6700\u5c0f\u5024\u306f\r\n\\[\r\n\\underline{\\dfrac{4}{3}}\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b63\u6570 \\(r\\) \u306b\u5bfe\u3057\u3066, \\(a _ 1 = 0\\) , \\(a _ 2 = r\\) \u3068\u304a\u304d, \u6570\u5217 \\(\\{ a _ n \\}\\) \u3092\u6b21\u306e\u6f38\u5316\u5f0f\u3067\u5b9a\u3081\u308b. \\[ a _ {n+1} = a _ n +r _ n ( … \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":[32],"tags":[142,13],"class_list":["post-31","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2011","tag-osaka_r","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/31","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=31"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/31\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=31"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=31"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=31"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}