{"id":1227,"date":"2015-08-13T23:13:05","date_gmt":"2015-08-13T14:13:05","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1227"},"modified":"2021-03-09T16:09:03","modified_gmt":"2021-03-09T07:09:03","slug":"tkr201504","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr201504\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2015\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\u6570\u5217 \\(\\{ p _ n \\}\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b. \r\n\\[\r\np _ 1 = 1 , \\ p _ 2 = 2 , \\ p _ {n+2} = \\dfrac{{p _ {n+1}^2 +1}}{p _ n} \\quad ( n = 1, 2, 3, \\cdots )\r\n\\]\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\dfrac{{p _ {n+1}}^2 +{p _ n}^2 +1}{p _ {n+1} p _ n}\\) \u304c \\(n\\) \u306b\u3088\u3089\u306a\u3044\u3053\u3068\u3092\u793a\u305b.\r\n

  2. (2)<\/strong>\u3000\u3059\u3079\u3066\u306e \\(n = 2, 3, 4, \\cdots\\) \u306b\u5bfe\u3057, \\(p _ {n+1} +p _ {n-1}\\) \u3092 \\(p _ n\\) \u306e\u307f\u3092\u4f7f\u3063\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u6570\u5217 \\(\\{ q _ n \\}\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b. \r\n\\[\r\nq _ 1 = 1 , \\ q _ 2 = 1 , \\ q _ {n+2} = q _ {n+1} +q _ n \\quad ( n = 1, 2, 3, \\cdots )\r\n\\]\r\n\u3059\u3079\u3066\u306e \\(n = 1, 2, 3, \\cdots\\) \u306b\u5bfe\u3057, \\(p _ n = q _ {2n-1}\\) \u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\(c _ n = \\dfrac{{p _ {n+1}}^2 +{p _ n}^2 +1}{p _ {n+1} p _ n}\\) \u3068\u304a\u304f.
    \r\n\u6761\u4ef6\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nc _ {n+1} & = \\dfrac{\\left( \\dfrac{{p _ {n+1}^2 +1}}{p _ n} \\right)^2 +{p _ {n+1}}^2 +1}{\\dfrac{{p _ {n+1}^2 +1}}{p _ n} \\cdot p _ {n+1}} \\\\\r\n& = \\dfrac{{p _ {n+1}}^4 +( {p _ n}^2 +2 ) {p _ {n+1}}^2 +{p _ n}^2 +1}{p _ {n+1} p _ n ( {p _ {n+1}}^2 +1 )} \\\\\r\n& = \\dfrac{{p _ {n+1}}^2 +{p _ n}^2 +1}{p _ {n+1} p _ n} = c _ n\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u3053\u308c\u3092\u7e70\u8fd4\u3057\u7528\u3044\u308c\u3070\r\n\\[\r\nc _ n = c _ 1 = \\dfrac{2^2 +1^2 +1}{2 \\cdot 1} = 3 \\quad ... [1]\r\n\\]\r\n\u3067, \\(n\\) \u306b\u3088\u3089\u305a\u4e00\u5b9a\u3068\u306a\u308b.<\/p>\r\n

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

    \u6761\u4ef6\u3088\u308a, \\({p _ {n+1}^2 +1 = p _ {n+2} p _ n}\\) \u306a\u306e\u3067, (1)<\/strong> \u306e\u7d50\u679c\u306b\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nc _ n = \\dfrac{p _ {n+2} p _ n +{p _ n}^2}{p _ {n+1} p _ n} & = \\dfrac{p _ {n+2} +p _ n}{p _ {n+1}} = 3 \\\\\r\n\\text{\u2234} \\quad p _ {n+2} +p _ n & = 3 p _ {n+1}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(n \\geqq 2\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\np _ {n+1} +p _ {n-1} = \\underline{3 p _ n}\r\n\\]\r\n

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

    \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066\r\n\\[\r\np _ n = q _ {2n-1} \\quad ... [ \\text{A} ]\r\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092, \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\\[\r\np _ 1 = q _ 1 = 1\r\n\\]\r\n\u306a\u306e\u3067, [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(n = 2\\) \u306e\u3068\u304d
      \r\n\\[\r\np _ 1 = q _ 3 = q _ 1 +q _ 2 = 2\r\n\\]\r\n\u306a\u306e\u3067, [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    3. 3*<\/strong>\u3000\\(n = k , k+1 \\ ( k \\geqq 1 )\\) \u306e\u3068\u304d\u306b [A] \u304c\u6210\u7acb\u3059\u308b, \u3059\u306a\u308f\u3061\r\n\\[\r\np _ k = q _ {2k-1} , \\ p _ {k+1} = q _ {2k+1}\r\n\\]\r\n\u3068\u4eee\u5b9a\u3059\u308b\u3068, (2)<\/strong> \u306e\u7d50\u679c\u3082\u7528\u3044\u3066\r\n\\[\\begin{align}\r\np _ {k+2} & = 3 p _ {k+1} -p _ k \\\\\r\n& = 3 q _ {2k+1} -q _ {2k-1} \\\\\r\n& = ( q _ {2k+2} -q _ {2k} ) +q _ {2k+1} +( q _ {2k} +q _ {2k-1} ) -q _ {2k-1} \\\\\r\n& = q _ {2k+2} +q _ {2k+1} \\\\\r\n& = q _ {2k+3}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(n = k+2\\) \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":"\u6570\u5217 \\(\\{ p _ n \\}\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b. \\[ p _ 1 = 1 , \\ p _ 2 = 2 , \\ p _ {n+2} = \\dfrac{{p _ {n+1}^2 +1}}{p _ n} \\quad … \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":[125],"tags":[139,137],"class_list":["post-1227","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2015","tag-tokyo_r","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1227","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=1227"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1227\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1227"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1227"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}