{"id":146,"date":"2011-12-01T14:39:28","date_gmt":"2011-12-01T05:39:28","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=146"},"modified":"2021-10-20T15:53:08","modified_gmt":"2021-10-20T06:53:08","slug":"ykr201005","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201005\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2010\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\u5404\u9805\u304c\u6b63\u306e\u5b9f\u6570\u3067\u3042\u308b\u6570\u5217 \\(\\left\\{ a _ n \\right\\}\\) \u304c \\(a _ 1 = 1\\) \u3068\u95a2\u4fc2\u5f0f\r\n\\[\r\na _ {n+1} - a _ n = \\sqrt{n} \\left( 1 + \\dfrac{1}{a _ n + a _ {n+1}} \\right) \\quad ( n = 1, 2, 3, \\cdots )\r\n\\]\r\n\u3092\u307f\u305f\u3057\u3066\u3044\u308b. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a _ n \\geqq \\sqrt{n} \\quad ( n = 1, 2, 3, \\cdots )\\) \u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\textstyle\\sum\\limits _ {k=1}^{n-1} \\sqrt{k} \\leqq \\dfrac{2}{3} \\left( n^{\\frac{3}{2}} -1 \\right) \\quad ( n = 2, 3, 4, \\cdots )\\) \u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(a _ n \\leqq \\dfrac{2}{3} n^{\\frac{3}{2}} + \\dfrac{1}{2} n - \\dfrac{1}{6} \\quad ( n = 1, 2, 3, \\cdots )\\) \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

    \u6761\u4ef6\u3088\u308a\r\n\\[\r\na _ {n} \\geqq 0 \\quad ( n = 1, 2, 3, \\cdots ) \\quad ... [1]\r\n\\]\r\n\u4e0e\u3048\u3089\u308c\u305f\u6f38\u5316\u5f0f\u306e\u53f3\u8fba\u306f\u6b63\u306a\u306e\u3067, \\(a _ {n+1} -a _ n \\gt 0\\) \u3059\u306a\u308f\u3061\r\n\\[\r\na _ {n+1} \\gt a _ n \\quad ( n = 1, 2, 3, \\cdots ) \\quad ... [2]\r\n\\]\r\n\u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066\r\n\\[\r\na _ n \\geqq \\sqrt{n} \\quad ... [\\text{A}]\r\n\\]\r\n\u304c\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\\(a _ 1 = 1 = \\sqrt{1}\\) \u306a\u306e\u3067, \u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(n = k \\ ( k \\geqq 1 )\\) \u306e\u3068\u304d, [A] \u304c\u6210\u7acb\u3059\u308b, \u3059\u306a\u308f\u3061\r\n\\[\r\na _ k \\geqq \\sqrt{k} \\quad ... [3]\r\n\\]\r\n\u3068\u4eee\u5b9a\u3059\u308b.\r\n\\[\\begin{align}\r\na _ {k+1} & = a _ k + \\sqrt{k} \\left( 1 + \\dfrac{1}{a _ k + a _ {k+1}} \\right) \\\\\r\n& \\gt a _ k + \\sqrt{k} \\quad \\left( \\ \\text{\u2235} \\ \\text{[1]\u3088\u308a} \\ \\dfrac{\\sqrt{k}}{a _ k + a _ {k+1}} \\gt 0 \\ \\right) \\\\\r\n& \\geqq 2 \\sqrt{k} \\quad ( \\ \\text{\u2235} \\ [3] \\ )\r\n\\end{align}\\]\r\n\u3053\u3053\u3067,\r\n\\[\r\n\\left( 2 \\sqrt{k} \\right)^2 -\\left( \\sqrt{k+1} \\right)^2 = 3k-1 \\gt 0\r\n\\]\r\n\u3059\u306a\u308f\u3061 \\(2 \\sqrt{k} \\gt \\sqrt{k+1}\\) \u306a\u306e\u3067\r\n\\[\r\na _ {k+1} \\gt \\sqrt{k+1}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(n = k+1\\) \u306e\u3068\u304d\u3082 [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      1*<\/strong> 2*<\/strong> \u3088\u308a, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066 [A] \u304c\u6210\u7acb\u3057, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

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

      \\(y = \\sqrt{x}\\) \u306e\u30b0\u30e9\u30d5\u306f, \\(x \\geqq 0\\) \u306b\u304a\u3044\u3066, \u4e0a\u306b\u51f8\u3067\u5358\u8abf\u5897\u52a0.<\/p>\r\n\"yokokoku2010_05_01\"\r\n

      \u4e0a\u56f3\u306e\u659c\u7dda\u90e8\u306e\u9762\u7a4d\u3068, \\(y=\\sqrt{x}\\) , \\(x\\) \u8ef8, \\(x = n\\) \u306b\u56f2\u307e\u308c\u305f\u90e8\u5206\u306e\u9762\u7a4d\u3092\u6bd4\u8f03\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {k=1}^{n-1} \\sqrt{k} & \\leqq \\displaystyle\\int _ 0^n \\sqrt{x} \\, dx = \\left[ \\dfrac{2}{3} x^{\\frac{3}{2}} \\right] _ 0^n \\\\\r\n& = \\dfrac{2}{3} \\left( n^{\\frac{3}{2}} -1 \\right) \\\\\r\n\\text{\u2234} \\quad \\textstyle\\sum\\limits _ {k=1}^{n-1} \\sqrt{k} & \\leqq \\dfrac{2}{3} \\left( n^{\\frac{3}{2}} -1 \\right)\r\n\\end{align}\\]\r\n

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

      (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(\\dfrac{\\sqrt{k}}{a _ k} \\leqq 1\\) ... [4] .
      \r\n\\(n \\geqq 2\\) \u306e\u3068\u304d\u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\na _ n & = a _ 1 +\\textstyle\\sum\\limits _ {k=1}^{n-1} \\left( a _ {k+1} -a _ k \\right) \\\\\r\n& = 1 + \\textstyle\\sum\\limits _ {k=1}^{n-1} \\left( \\sqrt{k} + \\dfrac{\\sqrt{k}}{a _ k + a _ {k+1}} \\right) \\\\\r\n& \\leqq 1 +\\textstyle\\sum\\limits _ {k=1}^{n-1} \\sqrt{k} +\\textstyle\\sum\\limits _ {k=1}^{n-1} \\dfrac{\\sqrt{k}}{2 a _ k} \\quad ( \\ \\text{\u2235} \\ [2] \\ ) \\\\\r\n& \\leqq 1 + \\dfrac{2}{3} \\left( n^{\\frac{3}{2}} -1 \\right) + \\dfrac{1}{2} ( n-1 ) \\quad ( \\ \\text{\u2235} \\ \\text{(2)\u306e\u7d50\u679c, [4]}\\ ) \\\\\r\n& = \\dfrac{2}{3} n^{\\frac{3}{2}} + \\dfrac{1}{2} n - \\dfrac{1}{6}\r\n\\end{align}\\]\r\n\u3053\u308c\u306f\r\n\\[\r\n\\dfrac{2}{3} \\cdot 1 + \\dfrac{1}{2} \\cdot 1 - \\dfrac{1}{6} = 1 = a _ 1\r\n\\]\r\n\u306a\u306e\u3067, \\(n=1\\) \u306e\u3068\u304d\u306b\u3082\u6210\u7acb\u3059\u308b.
      \r\n\u3088\u3063\u3066\r\n\\[\r\n\\underline{a _ n \\leqq \\dfrac{2}{3} n^{\\frac{3}{2}} + \\dfrac{1}{2} n - \\dfrac{1}{6}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5404\u9805\u304c\u6b63\u306e\u5b9f\u6570\u3067\u3042\u308b\u6570\u5217 \\(\\left\\{ a _ n \\right\\}\\) \u304c \\(a _ 1 = 1\\) \u3068\u95a2\u4fc2\u5f0f \\[ a _ {n+1} - a _ n = \\sqrt{n} \\left( 1 + \\dfrac … \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":[43],"tags":[14,9],"class_list":["post-146","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2010","tag-14","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/146","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=146"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/146\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=146"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=146"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=146"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}