{"id":253,"date":"2012-01-07T23:02:48","date_gmt":"2012-01-07T14:02:48","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=253"},"modified":"2021-03-16T17:12:48","modified_gmt":"2021-03-16T08:12:48","slug":"tkr200805","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr200805\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2008\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057, \\(\\dfrac{10^n -1}{9} =\\overbrace{111 \\cdots 111}^{n} = \\fbox{$n$}\\) \u3067\u8868\u3059.\r\n\u305f\u3068\u3048\u3070, \\(\\fbox{$1$}=1\\) , \\(\\fbox{$2$}=11\\) , \\(\\fbox{$3$}=111\\) \u3067\u3042\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(m\\) \u3092 \\(0\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b. \\(\\fbox{$3^m$}\\) \u306f \\(3^m\\) \u3067\u5272\u308a\u5207\u308c\u308b\u304c, \\(3^{m+1}\\) \u3067\u306f\u5272\u308a\u5207\u308c\u306a\u3044\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(n\\) \u304c \\(27\\) \u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u304c, \\(\\fbox{$n$}\\) \u304c \\(27\\) \u3067\u5272\u308a\u5207\u308c\u308b\u305f\u3081\u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\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

    \\(f(n) =\\dfrac{10^n -1}{9}\\) \u3068\u304a\u304f.<\/p>\r\n

      \r\n
    1. [A]\uff1a\u300e \\(f(3^m)\\) \u306f \\(3^m\\) \u3067\u5272\u308a\u5207\u308c\u308b\u304c, \\(3^{m+1}\\) \u3067\u306f\u5272\u308a\u5207\u308c\u306a\u3044\u300f<\/li>\r\n<\/ol>\r\n[A] \u304c \\(0\\) \u4ee5\u4e0a\u306e\u6574\u6570 \\(m\\) \u306b\u3064\u3044\u3066\u6210\u7acb\u3059\u308b\u3053\u3068\u3092, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3067\u793a\u3059.<\/p>\r\n
        \r\n

      1. 1*<\/strong>\u3000\\(m=0\\) \u306e\u3068\u304d\r\n\\[\r\nf(3^0) =f(1) =1\r\n\\]\r\n\u3053\u308c\u306f, \\(3^0 =1\\) \u3067\u5272\u308a\u5207\u308c\u308b\u304c, \\(3^1 =3\\) \u3067\u306f\u5272\u308a\u5207\u308c\u306a\u3044.
        \r\n\u3057\u305f\u304c\u3063\u3066, \\(m=0\\) \u306e\u3068\u304d, [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

      2. 2*<\/strong>\u3000\\(m=k \\ ( k \\geqq 0 )\\) \u306e\u3068\u304d, [A] \u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nf( 3^{k+1}) & = \\dfrac{10^{3 \\cdot 3^k} -1}{9} \\\\\r\n& = \\dfrac{\\left( 10^{3^k} -1 \\right) \\left( 10^{2 \\cdot 3^k} +10^{3^k} +1 \\right)}{9} \\\\\r\n& = f( 3^k ) \\underline{\\left( 10^{2 \\cdot 3^k} +10^{3^k} +1 \\right)} _ {[1]}\r\n\\end{align}\\]\r\n\u4e0b\u7dda\u90e8 [1] \u306b\u3064\u3044\u3066, \\(\\mod 3\\) \u3068\u3059\u308b\u3068\r\n\\[\r\n[1] \\equiv 1^{2 \\cdot 3^k} +1^{3^k} +1 \\equiv 0\r\n\\]\r\n\\(\\mod 9\\) \u3068\u3059\u308b\u3068\r\n\\[\r\n[1] \\equiv 1^{2 \\cdot 3^k} +1^{3^k} +1 \\equiv 3\r\n\\]\r\n\u306a\u306e\u3067, [1] \u306f \\(3\\) \u3067\u5272\u308a\u5207\u308c\u308b\u304c, \\(9\\) \u3067\u306f\u5272\u308a\u5207\u308c\u306a\u3044.
        \r\n\u4eee\u5b9a\u3088\u308a, \\(f(3^k)\\) \u306f, \\(3^k\\) \u3067\u306f\u5272\u308a\u5207\u308c\u308b\u304c, \\(3^{k+1}\\) \u3067\u306f\u5272\u308a\u5207\u308c\u306a\u3044\u306e\u3067,\r\n\\(f(3^{k+1})\\) \u306f, \\(3^{k+1}\\) \u3067\u306f\u5272\u308a\u5207\u308c\u308b\u304c, \\(3^{k+2}\\) \u3067\u306f\u5272\u308a\u5207\u308c\u306a\u3044.
        \r\n\u3057\u305f\u304c\u3063\u3066, \\(m=k+1\\) \u306e\u3068\u304d\u3082 [A] \u304c\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

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

          \r\n
        1. [P]\uff1a\u300e \\(n\\) \u304c \\(27\\) \u3067\u5272\u308a\u5207\u308c\u308b\u300f<\/p><\/li>\r\n
        2. [Q]\uff1a\u300e \\(f(n)\\) \u304c \\(27\\) \u3067\u5272\u308a\u5207\u308c\u308b\u300f<\/p><\/li>\r\n

        3. 1*<\/strong>\u3000\\(\\text{[P]} \\Longrightarrow \\text{[Q]}\\) \u306e\u8a3c\u660e
          \r\n\\(n =27 i\\) \uff08 \\(i\\) \u306f \\(0\\) \u4ee5\u4e0a\u306e\u6574\u6570\uff09\u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf(27i) & = \\dfrac{10^{27i}-1}{9} \\\\\r\n& = \\dfrac{10^{27}-1}{9} \\left\\{ 10^{27(i-1)} +10^{27(i-2)} +\\cdots +1 \\right\\} \\\\\r\n& = f( 3^3 ) \\left\\{ 10^{27(i-1)} +10^{27(i-2)} +\\cdots +1 \\right\\}\r\n\\end{align}\\]\r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(f(3^3)\\) \u306f \\(3^3 =27\\) \u3067\u5272\u308a\u5207\u308c\u308b\u306e\u3067, \\(f(27i)\\) \u306f27\u3067\u5272\u308a\u5207\u308c\u308b.<\/p><\/li>\r\n

        4. 2*<\/strong> \\(\\text{[Q]} \\Longrightarrow \\text{[P]}\\) \u306e\u8a3c\u660e
          \r\n\\(f(n)\\) \u306f \\(1\\) \u304c \\(n\\) \u500b\u4e26\u3093\u3060\u6570\u306a\u306e\u3067,
          \r\n\\(f(n) =27j\\) \uff08 \\(j\\) \u306f \\(0\\) \u4ee5\u4e0a\u306e\u6574\u6570\uff09\u306a\u3089\u3070, \\(n =27j\\) .<\/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":"\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057, \\(\\dfrac{10^n -1}{9} =\\overbrace{111 \\cdots 111}^{n} = \\fbox{$n$}\\) \u3067\u8868\u3059. \u305f\u3068\u3048\u3070, \\(\\fbox{$1$}=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":[17],"tags":[139,16],"class_list":["post-253","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2008","tag-tokyo_r","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/253","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=253"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/253\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=253"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=253"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=253"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}