{"id":920,"date":"2013-09-04T21:37:31","date_gmt":"2013-09-04T12:37:31","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=920"},"modified":"2021-09-30T00:01:30","modified_gmt":"2021-09-29T15:01:30","slug":"kbr201304","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr201304\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2013\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\\(3\\) \u3064\u306e\u6570\u5217 \\(\\{ a _ n \\} , \\{ b _ n \\} , \\{ c _ n \\}\\) \u304c\r\n\\[\\begin{align}\r\na _ {n+1} & = -b _ n -c _ n \\quad ( n = 1 , 2 , 3 , \\cdots ) \\\\\r\nb _ {n+1} & = -c _ n -a _ n \\quad ( n = 1 , 2 , 3 , \\cdots ) \\\\\r\nc _ {n+1} & = -a _ n -b _ n \\quad ( n = 1 , 2 , 3 , \\cdots )\r\n\\end{align}\\]\r\n\u304a\u3088\u3073 \\(a _ 1 = a\\) , \\(b _ 1 = b\\) , \\(c _ 1 = c\\) \u3092\u6e80\u305f\u3059\u3068\u3059\u308b. \u305f\u3060\u3057, \\(a , b , c\\) \u306f\u5b9a\u6570\u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(p _ n = a _ n +b _ n +c _ n \\ ( n = 1 , 2 , 3 , \\cdots )\\) \u3067\u4e0e\u3048\u3089\u308c\u308b\u6570\u5217 \\(\\{ p _ n \\}\\) \u306e\u521d\u9805\u304b\u3089\u7b2c \\(n\\) \u9805\u307e\u3067\u306e\u548c \\(S _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u6570\u5217 \\(\\{ a _ n \\} , \\{ b _ n \\} , \\{ c _ n \\}\\) \u306e\u4e00\u822c\u9805\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(q _ n = (-1)^n \\left\\{ (a _ n)^2 +(b _ n)^2 +(c _ n)^2 \\right\\} \\ ( n = 1 , 2 , 3 , \\cdots )\\) \u3067\u4e0e\u3048\u3089\u308c\u308b\u6570\u5217 \\(\\{ q _ n \\}\\) \u306e\u521d\u9805\u304b\u3089\u7b2c \\(2n\\) \u9805\u307e\u3067\u306e\u548c\u3092 \\(T _ n\\) \u3068\u3059\u308b. \\(a+b+c\\) \u304c\u5947\u6570\u3067\u3042\u308c\u3070, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066 \\(T _ n\\) \u304c\u6b63\u306e\u5947\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \u4e0e\u3048\u3089\u308c\u305f \\(3\\) \u5f0f\u3092\u8fba\u3005\u52a0\u3048\u308b\u3068\r\n\\[\r\np _ {n+1} = -2 p _ n\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6570\u5217 \\(\\{ p _ n \\}\\) \u306f, \u521d\u9805 \\(p _ 1 = a+b+c\\) , \u516c\u6bd4 \\(-2\\) \u306e\u7b49\u6bd4\u6570\u5217\u3067\u3042\u308a,\r\n\\[\r\np _ n = (-2)^{n-1} (a+b+c)\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u548c \\(S _ n\\) \u306f\r\n\\[\\begin{align}\r\nS _ n & = (a+b+c) \\dfrac{1 -(-2)^n}{1 -(-2)} \\\\\r\n& = \\underline{\\dfrac{a+b+c}{3} \\left\\{ 1 -(-2)^n \\right\\}}\r\n\\end{align}\\]\r\n

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

    \u6761\u4ef6\u3088\u308a\r\n\\[\\begin{gather}\r\na _ {n+1} = -p _ {n} +a _ n \\\\\r\n\\text{\u2234} \\quad a _ {n+1} -a _ n = -p _ {n}\r\n\\end{gather}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(n \\geqq 2\\) \u306b\u5bfe\u3057\u3066\r\n\\[\\begin{align}\r\na _ n & = a _ 1 + \\textstyle\\sum\\limits _ {k=1}^{n-1} (-p _ k) \\\\\r\n& = a -S _ {n-1} \\\\\r\n& = \\underline{a -\\dfrac{a+b+c}{3} \\left\\{ 1 -(-2)^{n-1} \\right\\}}\r\n\\end{align}\\]\r\n\u3053\u308c\u306f, \\(n=1\\) \u306e\u3068\u304d\u3082\u6210\u7acb\u3059\u308b.
    \r\n\u540c\u69d8\u306b\u8003\u3048\u308c\u3070\r\n\\[\\begin{align}\r\nb _ n & = \\underline{b -\\dfrac{a+b+c}{3} \\left\\{ 1 -(-2)^{n-1} \\right\\}} , \\\\\r\nc _ n & = \\underline{c -\\dfrac{a+b+c}{3} \\left\\{ 1 -(-2)^{n-1} \\right\\}}\r\n\\end{align}\\]\r\n

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

    \u307e\u305a\r\n\\[\r\nq _ {n} = (-1)^{n+1} \\left\\{ (p _ n)^2 -2 ( a _ n b _ n +b _ n c _ n +c _ n a _ n ) \\right\\}\r\n\\]\r\n\u306a\u306e\u3067, \u300c \\(p _ n\\) \u3068 \\(q _ n\\) \u306e\u5947\u5076\u306f\u4e00\u81f4\u3059\u308b. \u300d ... [1]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nq _ {2n-1} +q _ {2n} & = -\\left\\{ (a _ {2n-1})^2 +(b _ {2n-1})^2 +(c _ {2n-1})^2 \\right\\} \\\\\r\n& \\hspace{2em} 2 \\left\\{ (a _ {2n-1})^2 +(b _ {2n-1})^2 +(c _ {2n-1})^2 \\right. \\\\\r\n& \\hspace{4em} \\left. + a _ {2n-1} b _ {2n-1} +b _ {2n-1} c _ {2n-1} +c _ {2n-1} a _ {2n-1} \\right\\} \\\\\r\n& = ( p _ {2n-1} )^2\r\n\\end{align}\\]\r\n\u6761\u4ef6\u3088\u308a, \\(p _ 1\\) \u306f\u5947\u6570\u306a\u306e\u3067\u300c \\(q _ {1} +q _ {2}\\) \u306f\u6b63\u306e\u5947\u6570\u3067\u3042\u308b. \u300d ... [2]\r\n\u3055\u3089\u306b, (1)<\/strong> \u306e\u7d4c\u904e\u3088\u308a, \\(p _ n \\ ( n \\geqq 2 )\\) \u306f\u5076\u6570\u306a\u306e\u3067, \u300c \\(q _ {2n-1} +q _ {2n} \\ ( n \\geqq 2 )\\) \u306f \\(0\\) \u4ee5\u4e0a\u306e\u5076\u6570\u3067\u3042\u308b. \u300d ... [3]\r\n\u3053\u308c\u3089\u3092\u7528\u3044\u3066,<\/p>\r\n

      \r\n
    1. [A] ... \u300c \u548c \\(T _ {2n}\\) \u306f\u6b63\u306e\u5947\u6570\u3067\u3042\u308b. \u300d<\/li>\r\n<\/ol>\r\n

      \u304c, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\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[2] \u3088\u308a, [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

      2. 2*<\/strong>\u3000\\(n=k\\) \u306e\u3068\u304d, [A] \u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\r\nT _ {2(k+1)} = T _ {2k} +\\underline{q _ {2k+1} +q _ {2k+2}} _ {[4]}\r\n\\]\r\n\u306b\u304a\u3044\u3066, \u4eee\u5b9a\u3088\u308a \\(T _ {2k}\\) \u306f\u6b63\u306e\u5947\u6570\u3067\u3042\u308a, \u4e0b\u7dda\u90e8 [4] \u306f [3] \u3088\u308a \\(0\\) \u4ee5\u4e0a\u306e\u5076\u6570\u3067\u3042\u308b.
        \r\n\u3057\u305f\u304c\u3063\u3066, \\(T _ {2(k+1)}\\) \u306f\u6b63\u306e\u5947\u6570\u3067\u3042\u308a, \\(n = 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","protected":false},"excerpt":{"rendered":"\\(3\\) \u3064\u306e\u6570\u5217 \\(\\{ a _ n \\} , \\{ b _ n \\} , \\{ c _ n \\}\\) \u304c \\[\\begin{align} a _ {n+1} & = -b _ n -c _ n \\quad ( n … \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":[102],"tags":[144,111],"class_list":["post-920","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2013","tag-tsukuba_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/920","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=920"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/920\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=920"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=920"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=920"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}