{"id":783,"date":"2013-05-08T22:37:23","date_gmt":"2013-05-08T13:37:23","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=783"},"modified":"2021-03-23T17:13:57","modified_gmt":"2021-03-23T08:13:57","slug":"kyr201302","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr201302\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb2013\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(N\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3057,\r\n\\(a _ n\\) \uff08 \\(n=1, 2, \\cdots\\) \uff09\u3092\u6b21\u306e\u6027\u8cea (i)<\/strong> , (ii)<\/strong> \u3092\u307f\u305f\u3059\u6570\u5217\u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (i)<\/strong>\u3000\\(a _ 1 = 2^N-3\\) ,<\/p><\/li>\r\n

  2. (ii)<\/strong>\u3000\\(n=1, 2, \\cdots\\) \u306b\u5bfe\u3057\u3066,
    \r\n\\(a _ n\\) \u304c\u5076\u6570\u306e\u3068\u304d \\(a _ {n+1} = \\dfrac{a _ n}{2}\\) , \\(a _ n\\) \u304c\u5947\u6570\u306e\u3068\u304d \\(a _ {n+1} = \\dfrac{a _ n-1}{2}\\) .<\/p><\/li>\r\n<\/ol>\r\n

    \u3053\u306e\u3068\u304d\u3069\u306e\u3088\u3046\u306a\u81ea\u7136\u6570 \\(M\\) \u306b\u5bfe\u3057\u3066\u3082\r\n\\[\r\n\\sum\\limits _ {n=1}^M a _ n \\leqq 2^{N+1} -N-5\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p>\r\n

    \r\n\r\n

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

    \\[\r\n\\sum\\limits _ {n=1}^M a _ n \\leqq 2^{N+1} -N-5 \\quad ... [ \\text{A} ]\r\n\\]\r\n\\(S _ M = \\sum\\limits _ {n=1}^M a _ n\\) , \\(T _ N = 2^{N+1}-N-5\\) \u3068\u304a\u304f.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(N=2\\) \u306e\u3068\u304d\r\n\\[\r\na _ 1 = 2^2-3 = 1 , \\quad a _ 2 = 0\r\n\\]\r\n\u306a\u306e\u3067, \\(n \\geqq 2\\) \u306b\u5bfe\u3057\u3066\u306f\r\n\\[\r\na _ n = 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nS _ M = 1\r\n\\]\r\n\u307e\u305f\r\n\\[\r\nT _ 2 = 2^3 -2 -5 = 1\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(N=2\\) \u306e\u3068\u304d[A]\u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(N \\geqq 3\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\na _ 1 & = 2^{N -3} , \\\\\r\na _ 2 & = \\dfrac{a _ 1-1}{2} = 2^{N-1} -2 , \\\\\r\na _ 3 & = \\dfrac{a _ 2}{2} = 2^{N-2} -1\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(2^K -1\\) \uff08 \\(K\\) \u306f\u81ea\u7136\u6570\uff09\u306f\u5e38\u306b\u5947\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089, \\(3 \\leqq n \\leqq N\\) \u306b\u3064\u3044\u3066\r\n\\[\r\na _ n = 2^{N-n+1} -1\r\n\\]\r\n\u3055\u3089\u306b, \\(a _ N = 1\\) \u306a\u306e\u3067\r\n\\[\r\na _ {N+1} = 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(n \\geqq N+1\\) \u306b\u3064\u3044\u3066\r\n\\[\r\na _ n = 0\r\n\\]\r\n\u4ee5\u4e0a\u3088\u308a, \\(S _ M\\) \u306f \\(M\\) \u306b\u3064\u3044\u3066\u5358\u8abf\u5897\u52a0\u3067\u3042\u308a, \\(M \\geqq N\\) \u306e\u3068\u304d\u306b\u6700\u5927\u5024\u3092\u3068\u308b\u304b\u3089\r\n\\[\\begin{align}\r\nS _ M & \\leqq S _ N \\\\\r\n& = ( 2^{N}-3 ) +( 2^{N-2}-2 ) +( 2^{N-2}-1 ) \\\\\r\n& \\qquad +\\cdots +( 2-1 ) \\\\\r\n& = 2 \\cdot \\dfrac{2^N -1}{2-1} -5 -(N-2) \\\\\r\n& = 2^{N+1} -N -5 \\\\\r\n& = T _ N\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(n \\geqq 3\\) \u306e\u3068\u304d\u3082[A]\u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(N\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3057, \\(a _ n\\) \uff08 \\(n=1, 2, \\cdots\\) \uff09\u3092\u6b21\u306e\u6027\u8cea (i) , (ii) \u3092\u307f\u305f\u3059\u6570\u5217\u3068\u3059\u308b. (i)\u3000\\(a _ 1 = 2^N-3\\) , (i … \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":[88],"tags":[140,111],"class_list":["post-783","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2013","tag-kyoto_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/783","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=783"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/783\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=783"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=783"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=783"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}