{"id":1979,"date":"2021-11-18T08:46:30","date_gmt":"2021-11-17T23:46:30","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1979"},"modified":"2021-11-18T12:58:25","modified_gmt":"2021-11-18T03:58:25","slug":"tok202103","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok202103\/","title":{"rendered":"\u6771\u5de5\u59272021\uff1a\u7b2c3\u554f"},"content":{"rendered":"
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\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \u4e8c\u9805\u4fc2\u6570\u306b\u95a2\u3059\u308b\u6b21\u306e\u7b49\u5f0f\u3092\u793a\u305b.\r\n\\[\r\nn {} _ {2n} \\text{C} {} _ {n} = (n+1) {} _ {2n} \\text{C} {} _ {n-1}\r\n\\]\r\n\u307e\u305f, \u3053\u308c\u3092\u7528\u3044\u3066 \\({} _ {2n} \\text{C} {} _ {n}\\) \u306f \\(n+1\\) \u306e\u500d\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066,\r\n\\[\r\na_n = \\dfrac{{} _ {2n} \\text{C} {} _ {n}}{n+1}\r\n\\]\r\n\u3068\u304a\u304f. \u3053\u306e\u3068\u304d, \\(n \\geqq 4\\) \u306a\u3089\u3070 \\(a_n \\gt n+2\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(a_n\\) \u304c\u7d20\u6570\u3068\u306a\u308b\u6b63\u306e\u6574\u6570 \\(n\\) \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

    \\[\\begin{align}\r\nn {} _ {2n} \\text{C}{} _ {n} & = n \\cdot \\dfrac{2n (2n-1) \\cdots (n+2) (n+1)}{n (n-1) (n-2) \\cdots 1} \\\\\r\n& = \\dfrac{2n (2n-1) \\cdots (n+2) (n+1)}{(n-1) (n-2)\\cdots 1} \\\\\r\n& = (n+1) \\cdot \\dfrac{2n (2n-1) \\cdots (n+2)}{(n-1) (n-2)\\cdots 1} \\\\\r\n& = (n+1) {} _ {2n} \\text{C}{} _ {n-1}\r\n\\end{align}\\]\r\n\\(n\\) \u3068 \\(n+1\\) \u306f\u4e92\u3044\u306b\u7d20\u306a\u306e\u3067, \\({} _ {2n} \\text{C}{} _ {n}\\) \u306f \\(n+1\\) \u3092\u7d04\u6570\u306b\u3082\u3064, \u3059\u306a\u308f\u3061 \\(n+1\\) \u306e\u500d\u6570\u3067\u3042\u308b.<\/p>\r\n

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

    \\(a_n \\gt n+2\\) ... [A] \u3067\u3042\u308b\u3053\u3068\u3092\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(n = 4\\) \u306e\u3068\u304d\r\n\\[\r\na_4 = \\dfrac{{} _ {8} \\text{C}{} _ {4}}{5} = 15 \\gt 4+2\r\n\\]\r\n\u306a\u306e\u3067, [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(n = k \\ ( k \\geqq 4 )\\) \u306e\u3068\u304d
      \r\n[A] \u304c\u6210\u7acb\u3059\u308b, \u3059\u306a\u308f\u3061\r\n\\[\\begin{align}\r\na_k & = \\dfrac{{} _ {2k} \\text{C}{} _ {k}}{k+1} \\\\\r\n& = \\dfrac{2k (2k-1) \\cdots (k+2)}{k (k-1) \\cdots 1} \\gt k+2 \\quad ... [1]\r\n\\end{align}\\]\r\n\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\na _ {k+1} & = \\dfrac{{} _ {2(k+1)} \\text{C}{} _ {k+1}}{k+2} \\\\\r\n& = \\dfrac{(2k+2) (2k+1)}{(k+2) (k+1)} \\cdot \\dfrac{ 2k (2k-1) \\cdots (k+2)}{k (k-1) \\cdots 1} \\\\\r\n& \\gt \\dfrac{2 (2k+1)}{k+2} \\cdot (k+2) \\quad ( \\text{\u2235} \\ [1] \\ ) \\\\\r\n& = 4k+2 \\gt (k+1) +2 \\quad ( \\text{\u2235} \\ k \\geqq 4 \\ )\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(n = k+1\\) \u306e\u3068\u304d\u3082 [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      1*<\/strong> 2*<\/strong> \u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n

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

      (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(a_n\\) \u306f\u6b63\u306e\u6574\u6570\u3067\u3042\u308b.
      \r\n(2)<\/strong> \u306e\u9014\u4e2d\u7d4c\u904e\u304b\u3089\r\n\\[\r\na _ {n+1} = \\dfrac{2 (2n+1)}{n+2} a _ {n}\r\n\\]\r\n\\(a_n\\) \u304c\u5408\u6210\u6570\u3067\u3042\u308b\u3068\u304d, \\(a_n\\) \u3068 \\(n+2\\) \u306e\u6700\u5927\u516c\u7d04\u6570\u3092 \\(m\\) \u3068\u3057\u3066\r\n\\[\r\na_n = mp \\ , \\ n+2 = mq\r\n\\]\r\n\u3068\u8868\u305b\u3066, (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a \\(p \\geqq 2\\) ... [2] , \\(q \\leqq n+2\\) .\r\n\\[\r\na _ {n+1} = \\dfrac{2 (2n+1)}{q} p\r\n\\]\r\n\\(a _ {n+1}\\) \u306f\u6574\u6570\u306a\u306e\u3067, \\(2 (2n+1) = rq \\ ( \\ r \\text{\u306f\u6574\u6570} )\\) \u3067\u3042\u308a, \\(2 (2n+1) \\gt q\\) \u3060\u304b\u3089 \\(r \\geqq 2\\) ... [3] .
      \r\n\u3086\u3048\u306b, [2] [3] \u3088\u308a, \\(a _ {n+1}\\) \u3082\u5408\u6210\u6570\u3068\u306a\u308b.\r\n\\[\\begin{align}\r\na_1 = \\dfrac{{} _ {2} \\text{C}{} _ {1}}{2} = 1 \\ , & \\quad a_2 = \\dfrac{{} _ {4} \\text{C}{} _ {2}}{3} = 2 \\ , \\\\\r\na_3 = \\dfrac{{} _ {6} \\text{C}{} _ {3}}{4} = 5 \\ , & \\quad a_4 = \\dfrac{{} _ {8} \\text{C}{} _ {4}}{5} = 14\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(n \\geqq 4\\) \u3067\u306f \\(a_n\\) \u306f\u7d20\u6570\u3067\u306f\u306a\u3044.
      \r\n\u3088\u3063\u3066, \u6c42\u3081\u308b \\(n\\) \u306f\r\n\\[\r\nn = \\underline{2 , 3}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \u4e8c\u9805\u4fc2\u6570\u306b\u95a2\u3059\u308b\u6b21\u306e\u7b49\u5f0f\u3092\u793a\u305b. \\[ n {} _ {2n} \\text{C} {} _ {n} = (n+1) {} _ {2n} \\text{C} { … \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":[173],"tags":[141,165],"class_list":["post-1979","post","type-post","status-publish","format-standard","hentry","category-toko_2021","tag-toko","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1979","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=1979"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1979\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1979"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1979"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1979"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}