{"id":1856,"date":"2021-09-05T21:01:46","date_gmt":"2021-09-05T12:01:46","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1856"},"modified":"2021-09-08T13:22:45","modified_gmt":"2021-09-08T04:22:45","slug":"ngr201604","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201604\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2016\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\u6b21\u306e\u554f\u306b\u7b54\u3048\u3088. \u305f\u3060\u3057 \\(2\\) \u6b21\u65b9\u7a0b\u5f0f\u306e\u91cd\u89e3\u306f \\(2\\) \u3064\u3068\u6570\u3048\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u6b21\u306e\u6761\u4ef6 (\uff0a)<\/strong> \u3092\u6e80\u305f\u3059\u6574\u6570 \\(a , b , c , d , e , f\\) \u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.\r\n\\[\r\n\\text{(\uff0a)} \\ \\left\\{ \\begin{array}{l} 2 \\text{ \u6b21\u65b9\u7a0b\u5f0f } x^2 +ax +b = 0 \\text{ \u306e } 2 \\text{ \u3064\u306e\u89e3\u304c } c , d \\text{ \u3067\u3042\u308b. } \\\\ 2 \\text{ \u6b21\u65b9\u7a0b\u5f0f } x^2 +cx +d = 0 \\text{ \u306e } 2 \\text{ \u3064\u306e\u89e3\u304c } e , f \\text{ \u3067\u3042\u308b. } \\\\ 2 \\text{ \u6b21\u65b9\u7a0b\u5f0f } x^2 +ex +f = 0 \\text{ \u306e } 2 \\text{ \u3064\u306e\u89e3\u304c } a , b \\text{ \u3067\u3042\u308b. } \\end{array} \\right.\r\n\\]<\/li>\r\n

  2. (2)<\/strong>\u3000\\(2\\) \u3064\u306e\u6570\u5217 \\(\\{ a _ n \\} , \\{ b _ n \\}\\) \u306f, \u6b21\u306e\u6761\u4ef6 (\uff0a\uff0a)<\/strong> \u3092\u6e80\u305f\u3059\u3068\u3059\u308b.<\/p>\r\n

      \r\n
    1. (\uff0a\uff0a)<\/strong>\u3000\u3059\u3079\u3066\u306e\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u3064\u3044\u3066, \\(a _ n , b _ n\\) \u306f\u6574\u6570\u3067\u3042\u308a, \\(2\\) \u6b21\u65b9\u7a0b\u5f0f \\(x^2 +a _ n x +b _ n = 0\\) \u306e \\(2\\) \u3064\u306e\u89e3\u304c \\(a _ {n+1} , b _ {n+1}\\) \u3067\u3042\u308b.<\/li>\r\n<\/ol><\/li>\r\n<\/ol>\r\n

      \u3000\u3053\u306e\u3068\u304d<\/p>\r\n

        \r\n

      1. (i)<\/strong>\u3000\u6b63\u306e\u6574\u6570 \\(m\\) \u3067, \\(| b _ m | = | b _ {m+1} | = | b _ {m+2} | = \\cdots\\) \u3068\u306a\u308b\u3082\u306e\u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

      2. (ii)<\/strong>\u3000\u6761\u4ef6 (\uff0a\uff0a)<\/strong> \u3092\u6e80\u305f\u3059\u6570\u5217 \\(\\{ a _ n \\} , \\{ b _ n \\}\\) \u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

        \r\n\r\n

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

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

        (\uff0a)<\/strong> \u306e\u5404\u5f0f\u306b\u3064\u3044\u3066, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\\begin{align}\r\na & = -c-d \\quad ... [1] \\ , \\ b = cd \\quad ... [2] \\ , \\\\\r\nc & = -e-f \\quad ... [3] \\ , \\ d = ef \\quad ... [4] \\ , \\\\\r\ne & = -a-b \\quad ... [5] \\ , \\ f = ab \\quad ... [6] \\ .\r\n\\end{align}\\]\r\n[5] , [6] \u3092 [3] , [4] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\nc & = a +b -ab \\quad ... [7] \\ , \\\\\r\nd & = -ab ( a+b ) \\quad ... [8] \\ .\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u3055\u3089\u306b [1] , [2] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\na & = - (a+b ) +ab +ab ( a+b ) \\quad ... [9] \\ , \\\\\r\nb & = - ab ( a+b ) ( a+b -ab ) \\quad ... [10] \\ .\r\n\\end{align}\\]\r\n

          \r\n

        1. 1*<\/strong>\u3000\\(b = 0\\) \u306e\u3068\u304d
          \r\n[10] \u306f\u6210\u7acb\u3057\u3066, [9] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\na & = -a \\\\\r\n\\text{\u2234} \\quad a & = 0 \\ .\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d, [5] \uff5e [8] \u3088\u308a\r\n\\[\r\nc = d = e = f = 0 \\ .\r\n\\]<\/li>\r\n

        2. 2*<\/strong>\u3000\\(b \\neq 0\\) \u306e\u3068\u304d
          \r\n[8] \u306e\u4e21\u8fba\u3092 \\(b\\) \u3067\u5272\u3063\u3066\r\n\\[\r\n-a ( a+b ) ( a +b -ab ) = 1 \\ .\r\n\\]\r\n\u5de6\u8fba\u306f \\(3\\) \u3064\u306e\u6574\u6570\u306e\u7a4d\u3067\u3042\u308a, \u305d\u308c\u305e\u308c \\(1 , -1\\) \u306e\u3044\u305a\u308c\u304b\u3067\u3042\u308b.<\/p>\r\n

            \r\n
          1. (\u3042)<\/strong>\u3000\\(a = 1\\) \u306e\u3068\u304d
            \r\n\\[\\begin{align}\r\n( b+1 ) \\cdot 1 & = -1 \\\\\r\n\\text{\u2234} \\quad b & = -2 \\ .\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d, [5] \uff5e [8] \u3088\u308a\r\n\\[\r\nc = e = 1 , \\quad d = f = -2 \\ .\r\n\\]<\/li>\r\n
          2. (\u3044)<\/strong>\u3000\\(a = -1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n( b-1 ) ( 2b -1 ) & = 1 \\\\\r\nb ( 2b -3 ) & = 0 \\\\\r\n\\text{\u2234} \\quad b & = \\dfrac{3}{2} \\quad ( \\ \\text{\u2235} \\quad b \\neq 0 \\ ) \\ .\r\n\\end{align}\\]\r\n\u3053\u308c\u306f\u6574\u6570\u3067\u306f\u306a\u3044\u306e\u3067, \u4e0d\u9069.<\/li>\r\n<\/ol><\/li>\r\n<\/ol>\r\n

            \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u7d44\u306f\r\n\\[\r\n( a , b , c , d , e , f ) = \\underline{( 0 , 0 , 0 , 0 , 0 , 0 ) , ( 1 , -2 , 1 , -2 , 1 , -2 )} \\ .\r\n\\]\r\n

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

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

            (\uff0a\uff0a)<\/strong> \u306e \\(2\\) \u6b21\u65b9\u7a0b\u5f0f\u306b\u3064\u3044\u3066, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\na_n = -a _ {n+1} -b _ {n+1} \\quad ... [11] \\ , \\quad b_n = a _ {n+1} b _ {n+1} \\quad ... [12] \\ .\r\n\\]\r\n[12] \u3088\u308a, \u300c \\(b_n \\neq 0\\) \u306a\u3089\u3070 \\(a _ {n+1} \\neq 0\\) \u304b\u3064 \\(b _ {n+1} \\neq 0\\)\u300d... [13] \u306a\u306e\u3067, \u6b21\u306e\u3088\u3046\u306b\u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

              \r\n

            1. 1*<\/strong>\u3000\\(b_n = 0 \\ ( n \\geqq 1 )\\) \u306e\u3068\u304d
              \r\n\\(| b_n | = 0\\) \u3068\u306a\u308b\u306e\u3067, \u6761\u4ef6\u3092\u6e80\u305f\u3059 \\(m (=1)\\) \u304c\u5b58\u5728\u3059\u308b.<\/p><\/li>\r\n

            2. 2*<\/strong>\u3000\u3042\u308b\u81ea\u7136\u6570 \\(N\\) \u306b\u3064\u3044\u3066, \\(b_n \\neq 0 \\ ( n \\geqq N )\\) \u3068\u306a\u308b\u3068\u304d
              \r\n[12] \u3088\u308a, \\(| b_n | = | a _ {n+1} | | b _ {n+1} |\\) \u3067, [13] \u3088\u308a, \\(| b_n | , | a _ {n+1} | , | b _ {n+1} |\\) \u306f\u3059\u3079\u3066\u6b63\u306e\u6574\u6570\u306a\u306e\u3067\r\n\\[\r\n| b_n | \\geqq | b _ {n+1} | \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6570\u5217 \\(\\{ b_n \\}\\) \u306f, \\(n \\geqq N\\) \u306b\u304a\u3044\u3066\u5358\u8abf\u6e1b\u5c11\u3059\u308b.
              \r\n\u6761\u4ef6\u3092\u6e80\u305f\u3059 \\(m\\) \u304c\u5b58\u5728\u3057\u306a\u3044\u3068\u4eee\u5b9a\u3059\u308b\u3068,
              \r\n\\(| b_n |\\) \u306f\u6574\u6570\u5024\u3092\u3068\u308b\u306e\u3067, \u3069\u3053\u307e\u3067\u3082\u5c0f\u3055\u304f\u306a\u3063\u3066\u3044\u304f\u304c, \u3053\u308c\u306f \\(| b_n | \\geqq 1\\) \u3067\u3042\u308b\u3053\u3068\u306b\u77db\u76fe\u3059\u308b.
              \r\n\u3086\u3048\u306b, \u6761\u4ef6\u3092\u6e80\u305f\u3059 \\(m\\) \u304c\u5b58\u5728\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

              \u4ee5\u4e0a\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

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

              (i)<\/strong> \u3068\u540c\u3058\u5834\u5408\u5206\u3051\u3092\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

                \r\n

              1. 1*<\/strong> \u306e\u3068\u304d
                \r\n[11] \u3088\u308a, \\(a _ {n+1} = -a_n\\) \u306a\u306e\u3067,
                \r\n\u6570\u5217 \\(\\{ a_n \\}\\) \u306f, \u521d\u9805 \\(a_1 = k \\ ( k \\ \\text{\u306f\u6574\u6570} )\\) , \u516c\u6bd4 \\(-1\\) \u306e\u7b49\u6bd4\u6570\u5217\u3067\u3042\u308a\r\n\\[\r\na_n = (-1)^{n-1} k \\ .\r\n\\]<\/li>\r\n

              2. 2*<\/strong> \u306e\u3068\u304d
                \r\n\\(| b_m | = \\ell\\) \u3068\u304a\u304f.
                \r\n\\(n \\geqq m\\) \u306b\u304a\u3044\u3066, \\(| a_n | = | a_ {n+1} |\\) \u306a\u306e\u3067, \\( n \\geqq m+1\\) \u306b\u304a\u3044\u3066\r\n\\[\r\n| a_n | = 1 \\quad ... [14] \\ .\r\n\\]\r\n\\(2\\) \u6b21\u65b9\u7a0b\u5f0f\u306e\u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\r\nD = {a_n}^2 -4 b_n \\geqq 0 \\ .\r\n\\]\r\n\u306a\u306e\u3067, [14] \u3088\u308a, \\(n \\geqq m+1\\) \u306b\u304a\u3044\u3066\r\n\\[\r\nb_n \\leqq \\dfrac{1}{4} \\ .\r\n\\]\r\n\u3064\u307e\u308a\r\n\\[\r\nb_n = -\\ell \\quad ... [15] \\ .\r\n\\]\r\n[12] [14] \u3088\u308a, \\(n \\geqq m+2\\) \u306b\u304a\u3044\u3066\r\n\\[\r\na_n = 1 \\quad ... [16] \\ .\r\n\\]\r\n[11] [15] [16] \u3088\u308a, \\(n \\geqq m+3\\) \u306b\u304a\u3044\u3066\r\n\\[\\begin{align}\r\n1 & = -1 +\\ell \\\\\r\n\\text{\u2234} \\quad \\ell & = 2 \\\\\r\n\\text{\u2234} \\quad ( a_n , b_n ) & = ( 1 , -2 ) \\ .\r\n\\end{align}\\]\r\n\\(( a _ {n+1} , b _ {n+1} ) = ( 1 , -2 )\\) \u3067\u3042\u308c\u3070, [11] [12] \u3088\u308a\r\n\\[\r\n( a_n , b_n ) = ( 1 , -2 ) \\ .\r\n\\]\r\n\u306a\u306e\u3067, \u3053\u308c\u3092\u7e70\u8fd4\u3057\u7528\u3044\u308c\u3070, \u7d50\u5c40 \\(n \\geqq 1\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\n( a_n , b_n ) = ( 1 , -2 ) \\ .\r\n\\]<\/li>\r\n<\/ol>\r\n

                \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u6570\u5217\u306e\u7d44\u306f\r\n\\[\r\n( a_n , b_n ) = \\underline{( 1 , -2 ) , \\left( (-1)^{n-1} k , 0 \\right) \\quad ( k \\text{\u306f\u6574\u6570} \\ )} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u554f\u306b\u7b54\u3048\u3088. \u305f\u3060\u3057 \\(2\\) \u6b21\u65b9\u7a0b\u5f0f\u306e\u91cd\u89e3\u306f \\(2\\) \u3064\u3068\u6570\u3048\u308b. (1)\u3000\u6b21\u306e\u6761\u4ef6 (\uff0a) \u3092\u6e80\u305f\u3059\u6574\u6570 \\(a , b , c , d , e , f\\) \u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088. \\[ \\text{(\uff0a) … \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":[156],"tags":[143,162],"class_list":["post-1856","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2016","tag-nagoya_r","tag-162"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1856","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=1856"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1856\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1856"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1856"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1856"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}