{"id":27,"date":"2011-11-25T21:39:30","date_gmt":"2011-11-25T12:39:30","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=27"},"modified":"2021-09-09T20:37:45","modified_gmt":"2021-09-09T11:37:45","slug":"osr201101","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr201101\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2011\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(a\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. O \u3092\u539f\u70b9\u3068\u3059\u308b\u5ea7\u6a19\u5e73\u9762\u4e0a\u3067\u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & -1 \\\\ 1 & a \\end{array} \\right)\\) \u306e\u8868\u3059 \\(1\\) \u6b21\u5909\u63db\u3092 \\(f\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(r \\gt 0\\) \u304a\u3088\u3073 \\(0 \\leqq \\theta \\lt 2 \\pi\\) \u3092\u7528\u3044\u3066 \\(A = \\left( \\begin{array}{cc} r \\cos \\theta & -r \\sin \\theta \\\\ r \\sin \\theta & r \\cos \\theta \\end{array} \\right)\\) \u3092\u8868\u3059\u3068\u304d, \\(r\\) , \\(\\cos \\theta\\) , \\(\\sin \\theta\\) \u3092 \\(a\\) \u3067\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u70b9 Q \\(( 1 , 0 )\\) \u306b\u5bfe\u3057\u3066, \u70b9 \\(\\text{Q} _ n \\ ( n = 1, 2, \\cdots )\\) \u3092 \\(\\text{Q} _ 1 = \\text{Q}\\) , \\(\\text{Q} _ {n+1} = f( \\text{Q} _ n )\\) \u3067\u5b9a\u3081\u308b.\r\n\u25b3\\(\\text{OQ} _ n\\text{Q} _ {n+1}\\) \u306e\u9762\u7a4d \\(S(n)\\) \u3092 \\(a\\) \u3068 \\(n\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(f\\) \u306b\u3088\u3063\u3066\u70b9 \\(( 2 , 7 )\\) \u306b\u79fb\u3055\u308c\u308b\u3082\u3068\u306e\u70b9 P \u306e \\(x\\) \u5ea7\u6a19\u306e\u5c0f\u6570\u7b2c\u4e00\u4f4d\u3092\u56db\u6368\u4e94\u5165\u3057\u3066\u5f97\u3089\u308c\u308b\u8fd1\u4f3c\u5024\u304c \\(2\\) \u3067\u3042\u308b\u3068\u3044\u3046.\r\n\u81ea\u7136\u6570 \\(a\\) \u306e\u5024\u3092\u6c42\u3081\u3088. \u307e\u305f\u3053\u306e\u3068\u304d \\(S(n) \\gt 10^{10}\\) \u3068\u306a\u308b\u6700\u5c0f\u306e \\(n\\) \u306e\u5024\u3092\u6c42\u3081\u3088. \u305f\u3060\u3057 \\(0.3 \\lt \\log _ {10} 2 \\lt 0.31\\) \u3092\u7528\u3044\u3066\u3088\u3044.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\[\r\nA = \\sqrt{a^2+1} \\left( \\begin{array}{cc} \\dfrac{a}{\\sqrt{a^2+1}} & -\\dfrac{1}{\\sqrt{a^2+1}} \\\\ \\dfrac{1}{\\sqrt{a^2+1}} & \\dfrac{a}{\\sqrt{a^2+1}} \\end{array} \\right)\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nr= \\underline{\\sqrt{a^2+1}} , \\ \\cos \\theta =\\underline{\\dfrac{a}{\\sqrt{a^2+1}}} , \\ \\sin \\theta =\\underline{\\dfrac{1}{\\sqrt{a^2+1}}}\n\\]\r\n

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

    (1)<\/strong> \u306e\u7d50\u679c\u304b\u3089, \u70b9 P \u306b\u5bfe\u3057\u3066, \u70b9 \\(\\text{P}' =f(\\text{P})\\) \u306f\r\n\\[\r\n\\overrightarrow{\\text{OP}'} = r \\overrightarrow{\\text{OP}} , \\ \\angle \\text{POP}'= \\theta\n\\]\r\n\u3092\u6e80\u305f\u3059.
    \r\n\\(\\text{OQ} _ 1=1\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\text{OQ} _ n & = r^{n-1} , \\ \\text{OQ} _ {n+1} = r^n , \\ \\angle \\text{Q} _ n\\text{OQ} _ {n+1} = \\theta \\\\\r\n\\text{\u2234} \\quad S(n) & =\\dfrac{1}{2} r^{2n-1} \\sin \\theta =\\dfrac{1}{2} \\left( \\sqrt{a^2+1} \\right)^{2n-1} \\cdot \\dfrac{1}{\\sqrt{a^2+1}} \\\\\r\n& = \\underline{\\dfrac{1}{2}(a^2+1)^{n-1}}\\\n\\end{align}\\]\r\n

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

    \u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\n\\left( \\begin{array}{cc} a & -1 \\\\ 1 & a \\end{array} \\right) \\left( \\begin{array}{c} x \\\\ y \\end{array} \\right) & = \\left( \\begin{array}{c} 2 \\\\ 7 \\end{array} \\right) \\\\\r\n\\text{\u2234} \\quad \\left( \\begin{array}{c} x \\\\ y \\end{array} \\right) & = \\dfrac{1}{a^2+1}\\left( \\begin{array}{cc} a & -1 \\\\ 1 & a \\end{array} \\right) \\left( \\begin{array}{c} 2 \\\\ 7 \\end{array} \\right) \\\\\r\n& = \\dfrac{1}{a^2+1} \\left( \\begin{array}{c} 2a+7 \\\\ 7a-2 \\end{array} \\right)\n\\end{align}\\]\r\n\u3053\u306e\u70b9\u306e \\(x\\) \u5ea7\u6a19\u3092\u56db\u6368\u4e94\u5165\u3059\u308b\u3068 \\(2\\) \u3068\u306a\u308b\u306e\u3067\r\n\\[\\begin{align}\r\n\\dfrac{3}{2} & \\leqq \\dfrac{2a+7}{a^2+1} \\lt \\dfrac{5}{2} \\\\\r\n3a^2+3 & \\leqq 4a+14 \\lt 5a^2+5 \\\\\r\n3a^2-4a-11 & \\leqq 0 \\ \\text{\u304b\u3064} \\ (5a-9)(a+1) \\gt 0 \\\\\r\n\\dfrac{2-\\sqrt{37}}{3} \\leqq a & \\leqq \\dfrac{2+\\sqrt{37}}{3} \\ \\text{\u304b\u3064} \\ a \\lt -1 , \\dfrac{9}{5} \\lt a\n\\end{align}\\]\r\n\\(a\\) \u306f\u6574\u6570\u3067\u3042\u308a, \\(6 \\lt \\sqrt{37} \\lt 7\\) \u306a\u306e\u3067,\r\n\\[\\begin{align}\r\n-1 \\leqq a \\leqq 2 \\ & \\text{\u304b\u3064} \\ a \\leqq -2 , 2 \\leqq a \\\\\r\n\\text{\u2234} \\quad a & = \\underline{2}\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d, (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nS(n) = \\dfrac{1}{2} \\cdot 5^{n-1} = \\dfrac{10^{n-1}}{2^n} \\gt 10^{10}\n\\]\r\n\u4e21\u8fba\u306b\u3064\u3044\u3066\u5e38\u7528\u5bfe\u6570\u3092\u3068\u308b\u3068\r\n\\[\\begin{align}\r\nn-1-n \\log _ {10} 2 & \\gt 10 \\\\\r\n\\text{\u2234} \\quad n & \\gt \\dfrac{11}{1 -\\log _ {10} 2}\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(0.3 \\lt \\log _ {10} 2 \\lt 0.31\\) \u3067\u3042\u308a, \\(\\dfrac{11}{1-0.31} =15.9 \\cdots\\) \u306a\u306e\u3067\r\n\\[\r\nn \\geqq 16\n\\]\r\n\u3086\u3048\u306b, \u6700\u5c0f\u306e \\(n\\) \u306f\r\n\\[\r\nn= \\underline{16}\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. O \u3092\u539f\u70b9\u3068\u3059\u308b\u5ea7\u6a19\u5e73\u9762\u4e0a\u3067\u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & -1 \\\\ 1 & a \\end{array} \\right)\\) \u306e\u8868\u3059 \\(1\\) … \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-27","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\/27","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=27"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/27\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=27"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=27"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=27"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}