{"id":1126,"date":"2015-06-22T10:00:31","date_gmt":"2015-06-22T01:00:31","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1126"},"modified":"2021-09-24T17:30:12","modified_gmt":"2021-09-24T08:30:12","slug":"tok201401","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201401\/","title":{"rendered":"\u6771\u5de5\u59272014\uff1a\u7b2c1\u554f"},"content":{"rendered":"
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\\(3\\) \u4ee5\u4e0a\u306e\u5947\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \\(a _ n\\) \u3068 \\(b _ n\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b.\r\n\\[\r\na _ n = \\dfrac{1}{6} \\textstyle\\sum\\limits _ {k=1}^{n-1} (k-1) k (k+1) , \\ \\ b _ n = \\dfrac{n^2-1}{8}\r\n\\]\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a _ n\\) \u3068 \\(b _ n\\) \u306f\u3069\u3061\u3089\u3082\u6574\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(a _ n -b _ n\\) \u306f \\(4\\) \u306e\u500d\u6570\u3067\u3042\u308b\u3053\u3068\u3092\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

    \\(3\\) \u3064\u306e\u9023\u7d9a\u3059\u308b\u6574\u6570\u306b\u306f, \\(2\\) \u306e\u500d\u6570\u3068 \\(3\\) \u306e\u500d\u6570\u304c\u5c11\u306a\u304f\u3068\u3082 \\(1\\) \u3064\u305a\u3064\u542b\u307e\u308c\u3066\u3044\u308b\u306e\u3067\r\n\\[\r\n(k-1) k (k+1) \\ \\text{\u306f} \\ 6 \\ \\text{\u306e\u500d\u6570\u3067\u3042\u308b. } \\quad ... [1]\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\((k-1) k (k+1)\\) \u306e\u548c\u3082, \\(6\\) \u306e\u500d\u6570\u3060\u304b\u3089, \\(a _ n\\) \u306f\u6574\u6570\u3067\u3042\u308b.
    \r\n\\(3\\) \u4ee5\u4e0a\u306e\u5947\u6570\u306f \\(4m \\pm 1 \\quad ( m \\text{\u306f\u81ea\u7136\u6570} )\\) \u3068\u8868\u305b\u308b\u306e\u3067\r\n\\[\r\nb _ n = \\dfrac{\\left( 4m \\pm 1 \\right)^2 -1}{8} = 2 m^2 \\pm m\r\n\\]\r\n\u3088\u3063\u3066, \\(b _ n\\) \u306f\u6574\u6570\u3067\u3042\u308b.<\/p>\r\n

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

    \\(f(k) = (k-2) (k-1) k (k+1)\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\na _ n & = \\dfrac{1}{6} \\textstyle\\sum\\limits _ {k=1}^{n-1} \\dfrac{1}{4} \\left( f(k+1) -f(k) \\right) \\\\\r\n& = \\dfrac{1}{24} \\left( f(n) - f(1) \\right) \\\\\r\n& = \\dfrac{1}{24} (n-2) (n-1) n (n+1)\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\na _ n - b _ n & = \\dfrac{1}{24} (n-1) (n+1) \\left\\{ n (n-2) -3 \\right\\} \\\\\r\n& = \\dfrac{1}{24} (n-3) (n-1) (n+1)^2\r\n\\end{align}\\]\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(n = 4m+1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\na _ n - b _ n & = \\dfrac{1}{24} (4m-2) 4m (4m+2)^2 \\\\\r\n& = \\dfrac{2}{3} \\underline{(2m-1) 2m (2m+1)^2} _ {[2]}\r\n\\end{align}\\]\r\n[1] \u3088\u308a, \u4e0b\u7dda\u90e8 [2] \u306f \\(6\\) \u306e\u500d\u6570\u306a\u306e\u3067, \\(a _ n - b _ n\\) \u306f \\(4\\) \u306e\u500d\u6570\u3067\u3042\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(n = 4m-1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\na _ n - b _ n & = \\dfrac{1}{24} (4m-4) (4m-2) (4m)^2 \\\\\r\n& = \\dfrac{2}{3} \\underline{(2m-2) (2m-1) (2m)^2} _ {[3]}\r\n\\end{align}\\]\r\n[1] \u3088\u308a, \u4e0b\u7dda\u90e8 [3] \u306f \\(6\\) \u306e\u500d\u6570\u306a\u306e\u3067, \\(a _ n - b _ n\\) \u306f \\(4\\) \u306e\u500d\u6570\u3067\u3042\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      1*<\/strong> 2*<\/strong>\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(3\\) \u4ee5\u4e0a\u306e\u5947\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \\(a _ n\\) \u3068 \\(b _ n\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b. \\[ a _ n = \\dfrac{1}{6} \\textstyle\\sum\\limits _ {k=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":[116],"tags":[141,112],"class_list":["post-1126","post","type-post","status-publish","format-standard","hentry","category-toko_2014","tag-toko","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1126","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=1126"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1126\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1126"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1126"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}