{"id":345,"date":"2012-03-24T20:33:58","date_gmt":"2012-03-24T11:33:58","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=345"},"modified":"2021-10-20T19:35:38","modified_gmt":"2021-10-20T10:35:38","slug":"ykr200805","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr200805\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2008\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\u6570\u5217 \\(\\left\\{ a _ n \\right\\}\\) \u3092\r\n\\[\r\na _ 1 = \\dfrac{1}{2} , \\ a _ {n+1} =1-{a _ n}^2 \\quad ( n =1, 2, 3, \\cdots )\r\n\\]\r\n\u3067\u5b9a\u3081\u308b. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(0 \\lt a _ {2n-1} \\leqq \\dfrac{1}{2}\\) , \\(\\dfrac{3}{4} \\leqq a _ {2n} \\lt 1\\) \uff08 \\(n =1, 2, 3, \\cdots\\) \uff09\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(x\\) \u304c \\(0 \\leqq x \\leqq \\dfrac{1}{2}\\) \u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \u95a2\u6570 \\(f(x) = 2x -x^3\\) \u306e\u3068\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u308b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(\\dfrac{2 _ {2n+1}}{a _ {2n-1}} \\leqq \\dfrac{7}{8}\\) \uff08 \\(n =1, 2, 3, \\cdots\\) \uff09\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\[\r\n0 \\lt a _ {2n-1} \\leqq \\dfrac{1}{2} , \\ \\dfrac{3}{4} \\leqq a _ {2n} \\lt 1 \\quad ... [\\text{A}]\r\n\\]\r\n\u304c\u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066\u6210\u7acb\u3059\u308b\u3053\u3068\u3092, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(n = 1\\) \u306e\u3068\u304d\r\n\\[\r\na _ 1 = \\dfrac{1}{2} , \\ a _ 2 = 1-\\left( \\dfrac{1}{2} \\right)^2 = \\dfrac{3}{4}\r\n\\]\r\n\u3086\u3048\u306b, [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(n = k\\) \uff08 \\(k\\) \u306f\u81ea\u7136\u6570\uff09\u306e\u3068\u304d, [A] \u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n1 -1^2 & \\lt a _ {2k+1} \\leqq 1 -\\left( \\dfrac{3}{4} \\right)^2 \\\\\r\n\\text{\u2234} \\quad 0 & \\lt a _ {2k+1} \\leqq \\dfrac{7}{16} <\\dfrac{1}{2}\r\n\\end{align}\\]\r\n\u3055\u3089\u306b\u3053\u308c\u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\n1 -\\left( \\dfrac{1}{2} \\right)^2 \\leqq a _ {2k+2} & \\lt 1 -0^2 \\\\\r\n\\text{\u2234} \\quad \\dfrac{3}{4} \\leqq a _ {2k+2} & \\lt 1\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \\(n = k+1\\) \u306e\u3068\u304d\u3082 [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      1*<\/strong> 2*<\/strong> \u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

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

      \\(0 \\leqq x \\leqq \\dfrac{1}{2}\\) \u306e\u3068\u304d\r\n\\[\r\nf'(x) = 2 -3x^2 \\geqq 2 -3 \\left( \\dfrac{1}{2} \\right)^2 = \\dfrac{5}{4} \\gt 0\r\n\\]\r\n\u306a\u306e\u3067, \\(f(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u3059\u308b.\r\n\u3088\u3063\u3066\r\n\\[\\begin{gather}\r\nf(0) \\leqq f(x) \\leqq f \\left( \\dfrac{1}{2} \\right) \\\\\r\n\\text{\u2234} \\quad \\underline{0 \\leqq f(x) \\leqq \\dfrac{7}{8}}\r\n\\end{gather}\\]\r\n

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

      \\[\\begin{align}\r\na _ {2n+1} & = 1-{a _ {2n}}^2 =1-\\left( 1-{a _ {2n-1}}^2 \\right)^2 \\\\\r\n& = 2 {a _ {2n-1}}^2 -{a _ {2n-1}}^4 \\\\\r\n\\text{\u2234} \\quad & \\dfrac{a _ {2n+1}}{a _ {2n-1}} = 2 a _ {2n-1} -{a _ {2n-1}}^3\r\n\\end{align}\\]\r\n(1)<\/strong> \u306e\u7d50\u679c\u304b\u3089 \\(0 \\lt a _ {2n-1} \\leqq \\dfrac{1}{2}\\) \u306a\u306e\u3067, (2)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n\\dfrac{a _ {2n+1}}{a _ {2n-1}} = f( a _ {2n-1} ) \\leqq \\dfrac{7}{8}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6570\u5217 \\(\\left\\{ a _ n \\right\\}\\) \u3092 \\[ a _ 1 = \\dfrac{1}{2} , \\ a _ {n+1} =1-{a _ n}^2 \\quad ( n =1, 2, 3, \\cdots … \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":[56],"tags":[16,9],"class_list":["post-345","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2008","tag-16","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/345","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=345"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/345\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=345"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=345"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=345"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}