{"id":703,"date":"2013-03-29T23:20:19","date_gmt":"2013-03-29T14:20:19","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=703"},"modified":"2021-10-04T19:15:59","modified_gmt":"2021-10-04T10:15:59","slug":"kbr200704","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr200704\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2007\uff1a\u7b2c4\u554f"},"content":{"rendered":"
    \r\n

  1. (1)<\/strong>\u3000\u4e00\u822c\u9805 \\(a _ n\\) \u304c \\(an^3 +bn^2 +cn\\) \u3067\u8868\u3055\u308c\u308b\u6570\u5217 \\(\\{ a _ n \\}\\) \u306b\u304a\u3044\u3066,\r\n\\[\r\nn^2 = a _ {n+1} -a _ n \\quad ( n = 1, 2, 3, \\cdots )\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3088\u3046\u306b, \u5b9a\u6570 \\(a , b , c\\) \u3092\u5b9a\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000(1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u3066, \\(\\textstyle\\sum\\limits _ {k=1}^n k^2 = \\dfrac{1}{6} n(n+1)(2n+1)\\) \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(1, 2, 3, \\cdots , n\\) \u306e\u76f8\u7570\u306a\u308b \\(2\\) \u6570\u306e\u7a4d\u306e\u3059\u3079\u3066\u306e\u548c\u3092 \\(S(n)\\) \u3068\u3059\u308b. \u305f\u3068\u3048\u3070, \\(S(3) = 1 \\times 2 +1 \\times 3 +2 \\times 3 = 11\\) \u3067\u3042\u308b. \\(S(n)\\) \u3092 \\(n\\) \u306e \\(4\\) \u6b21\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\[\\begin{align}\r\na _ {n+1} -a _ n & = a(n+1)^3 +b(n+1)^2 +c(n+1) \\\\\r\n& \\qquad -an^3-bn^2-cn \\\\\r\n& = 3a n^2 +(3a+2b) n+(a+b+c)\r\n\\end{align}\\]\r\n\\(n^2 = a _ {n+1} -a _ n\\) \u306a\u306e\u3067, \u4fc2\u6570\u3092\u6bd4\u8f03\u3057\u3066\r\n\\[\\begin{gather}\r\n3a = 1 , \\ 3a+2b = 0 , \\ a+b+c = 0 \\\\\r\n\\text{\u2234} \\quad ( a , b , c ) = \\underline{\\left( \\dfrac{1}{3} , \\ -\\dfrac{1}{2} , \\ \\dfrac{1}{6} \\right)}\r\n\\end{gather}\\]\r\n

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

    \\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {k=1}^n k^2 & = \\textstyle\\sum\\limits _ {k=1}^n \\left( a _ {k+1} -a _ k \\right) \\\\\r\n& = a _ {n+1} -a _ 1 \\\\\r\n& = \\dfrac{(n+1)^3}{3} -\\dfrac{(n+1)^2}{2} +\\dfrac{n+1}{6} -0 \\\\\r\n& = \\dfrac{1}{6} (n+1) \\left\\{ 2(n+1)^2 -3(n+1) +1 \\right\\} \\\\\r\n& = \\dfrac{1}{6} (n+1)(2n^2+n) \\\\\r\n& = \\dfrac{1}{6} n(n+1)(2n+1)\r\n\\end{align}\\]\r\n

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

    \\[\r\n2 S(n) = \\left( 1 +2 + \\cdots + n \\right)^2 -\\left( 1^2 +2^2 + \\cdots + n^2 \\right)\r\n\\]\r\n\u3067\u3042\u308b\u3053\u3068\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nS(n) & = \\dfrac{1}{2} \\left\\{ \\dfrac{1}{4} n^2 (n+1)^2 -\\dfrac{1}{6} n(n+1)(2n+1) \\right\\} \\\\\r\n& = \\dfrac{1}{24} n(n+1) \\left\\{ 3n(n+1) -2(2n+1) \\right\\} \\\\\r\n& = \\dfrac{1}{24} n(n+1)(3n^2-n-2) \\\\\r\n& = \\underline{\\dfrac{1}{24} n(n-1)(n+1)(3n+2)}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"(1)\u3000\u4e00\u822c\u9805 \\(a _ n\\) \u304c \\(an^3 +bn^2 +cn\\) \u3067\u8868\u3055\u308c\u308b\u6570\u5217 \\(\\{ a _ n \\}\\) \u306b\u304a\u3044\u3066, \\[ n^2 = a _ {n+1} -a _ n \\quad ( n = 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":[101],"tags":[144,109],"class_list":["post-703","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2007","tag-tsukuba_r","tag-109"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/703","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=703"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/703\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=703"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=703"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=703"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}