{"id":1178,"date":"2015-07-21T23:52:31","date_gmt":"2015-07-21T14:52:31","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1178"},"modified":"2021-09-29T23:04:42","modified_gmt":"2021-09-29T14:04:42","slug":"tbr201404","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tbr201404\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2014\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\u5e73\u9762\u4e0a\u306e\u76f4\u7dda \\(\\ell\\) \u306b\u540c\u3058\u5074\u3067\u63a5\u3059\u308b \\(2\\) \u3064\u306e\u5186 \\(C _ 1 , C _ 2\\) \u304c\u3042\u308a, \\(C _ 1\\) \u3068 \\(C _ 2\\) \u3082\u4e92\u3044\u306b\u5916\u63a5\u3057\u3066\u3044\u308b.\r\n\\(\\ell , C _ 1 , C _ 2\\) \u3067\u56f2\u307e\u308c\u305f\u9818\u57df\u5185\u306b, \u3053\u308c\u3089 \\(3\\) \u3064\u3068\u4e92\u3044\u306b\u63a5\u3059\u308b\u5186 \\(C _ 3\\) \u3092\u4f5c\u308b. \u540c\u69d8\u306b \\(\\ell , C _ n , C _ {n+1} \\ ( n = 1, 2, 3, \\cdots )\\) \u3067\u56f2\u307e\u308c\u305f\u9818\u57df\u5185\u306b\u3042\u308a, \u3053\u308c\u3089 \\(3\\) \u3064\u3068\u4e92\u3044\u306b\u63a5\u3059\u308b\u5186 \\(C _ {n+2}\\) \u3068\u3059\u308b.\r\n\u5186 \\(C _ n\\) \u306e\u534a\u5f84\u3092 \\(r _ n\\) \u3068\u3057, \\(x _ n = \\dfrac{1}{\\sqrt{r _ n}}\\) \u3068\u304a\u304f.\r\n\u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. \u305f\u3060\u3057, \\(r _ 1 = 16\\) , \\(r _ 2 = 9\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\ell\\) \u304c \\(C _ 1 , C _ 2 , C _ 3\\) \u3068\u63a5\u3059\u308b\u70b9\u3092, \u305d\u308c\u305e\u308c \\(A _ 1 , A _ 2 , A _ 3\\) \u3068\u304a\u304f. \u7dda\u5206 \\(A _ 1 A _ 2 , A _ 1 A _ 3 , A _ 2 A _ 3\\) \u306e\u9577\u3055\u304a\u3088\u3073 \\(r _ 3\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u3042\u308b\u5b9a\u6570 \\(a , b\\) \u306b\u5bfe\u3057\u3066 \\(x _ {n+2} = a x _ {n+1} +b x _ n \\ ( n = 1, 2, 3, \\cdots )\\) \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b. \\(a , b\\) \u306e\u5024\u3082\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000(2)<\/strong> \u3067\u6c42\u3081\u305f \\(a , b\\) \u306b\u5bfe\u3057\u3066, \\(2\\) \u6b21\u65b9\u7a0b\u5f0f \\(t^2 = at +b\\) \u306e\u89e3\u3092 \\(\\alpha , \\beta \\ ( \\alpha \\gt \\beta )\\) \u3068\u3059\u308b. \\(x _ 1 = c {\\alpha}^2 +d {\\beta}^2\\) \u3092\u6e80\u305f\u3059\u6709\u7406\u6570 \\(c , d\\) \u306e\u5024\u3092\u6c42\u3081\u3088. \u305f\u3060\u3057, \\(\\sqrt{5}\\) \u304c\u7121\u7406\u6570\u3067\u3042\u308b\u3053\u3068\u306f\u8a3c\u660e\u306a\u3057\u3067\u7528\u3044\u3066\u3088\u3044.<\/p><\/li>\r\n

  4. (4)<\/strong>\u3000(3)<\/strong> \u306e \\(c , d , \\alpha , \\beta\\) \u306b\u5bfe\u3057\u3066,\r\n\\[\r\nx _ n = c {\\alpha}^{n+1} +d {\\beta}^{n+1} \\quad ( n = 1, 2, 3, \\cdots )\r\n\\]\r\n\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3057, \u6570\u5217 \\(\\{ r _ n \\}\\) \u306e\u4e00\u822c\u9805\u3092 \\(\\alpha , \\beta\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n\"tbr20140401\"\r\n

    \r\n\r\n

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

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

    \\[\\begin{align}\r\n\\text{A} _ 1 \\text{A} _ 2 & = \\sqrt{( r _ 1 +r _ 2 )^2 -( r _ 1 -r _ 2 )^2} \\\\\r\n& = 2 \\sqrt{r _ 1 r _ 2} = \\underline{24}\r\n\\end{align}\\]\r\n\u540c\u69d8\u306b\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n\\text{A} _ 1 \\text{A} _ 3 & = 2 \\sqrt{r _ 1 r _ 3} = 8 \\sqrt{r _ 3} , \\\\\r\n\\text{A} _ 2 \\text{A} _ 3 & = 2 \\sqrt{r _ 2 r _ 3} = 6 \\sqrt{r _ 3}\r\n\\end{align}\\]\r\n\\(\\text{A} _ 1 \\text{A} _ 2 = \\text{A} _ 1 \\text{A} _ 3 +\\text{A} _ 2 \\text{A} _ 3\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n8 \\sqrt{r _ 3} +6 \\sqrt{r _ 3} & = 24 \\\\\r\n\\sqrt{r _ 3} & = \\dfrac{12}{7} \\\\\r\n\\text{\u2234} \\quad r _ 3 & = \\underline{\\dfrac{144}{49}}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\text{A} _ 1 \\text{A} _ 3 = \\underline{\\dfrac{96}{7}} , \\ \\text{A} _ 2 \\text{A} _ 3 = \\underline{\\dfrac{72}{7}}\r\n\\]\r\n

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

    \\(C _ n , C _ {n+1} , C _ {n+2}\\) \u306b\u3064\u3044\u3066, (1)<\/strong> \u3068\u540c\u69d8\u306b\u8003\u3048\u308c\u3070\r\n\\[\\begin{align}\r\n2 \\sqrt{r _ n r _ {n+1}} & = 2 \\sqrt{r _ n r _ {n+2}} +2 \\sqrt{r _ {n+1} r _ {n+2}} \\\\\r\n\\dfrac{1}{\\sqrt{r _ {n+2}}} & = \\dfrac{1}{\\sqrt{r _ {n+1}}} +\\dfrac{1}{\\sqrt{r _ n}} \\\\\r\n\\text{\u2234} \\quad x _ {n+2} & = x _ {n+1} +x _ n\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\na = b = \\underline{1}\r\n\\]\r\n

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

    \\(\\alpha , \\beta \\ ( \\alpha \\gt \\beta )\\) \u306f \u65b9\u7a0b\u5f0f \\(t^2 -t -1 = 0\\) \u306e\u89e3\u306a\u306e\u3067\r\n\\[\r\n\\alpha = \\dfrac{1 +\\sqrt{5}}{2} , \\ \\beta = \\dfrac{1 -\\sqrt{5}}{2}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n{\\alpha}^2 = \\alpha +1 = \\dfrac{3 +\\sqrt{5}}{2} , \\\\\r\n{\\beta}^2 = \\beta +1 = \\dfrac{3 -\\sqrt{5}}{2}\r\n\\end{align}\\]\r\n\\(x _ 1 = \\dfrac{1}{4}\\) \u3068, \u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nc {\\alpha}^2 +d {\\beta}^2 & = \\dfrac{3}{2} (c+d) +\\dfrac{\\sqrt{5}}{2} (c-d) = \\dfrac{1}{4}\r\n\\end{align}\\]\r\n\\(c , d\\) \u306f\u6709\u7406\u6570, \\(\\sqrt{5}\\) \u306f\u7121\u7406\u6570\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\dfrac{3 (c+d)}{2} = \\dfrac{1}{4} & , \\ \\dfrac{c-d}{2} = 0 \\\\\r\n\\text{\u2234} \\quad c = d & = \\underline{\\dfrac{1}{12}}\r\n\\end{align}\\]\r\n

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

    \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066\r\n\\[\r\nx _ n = c {\\alpha}^{n+1} +d {\\beta}^{n+1} \\quad ... [ \\text{A} ]\r\n\\]\r\n\u304c\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(3)<\/strong> \u306e\u7d50\u679c\u3088\u308a, [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(n = 2\\) \u306e\u3068\u304d
      \r\n\\(x _ 2 = c' {\\alpha}^3 +d' {\\beta}^3\\) \u3092\u307f\u305f\u3059 \u6709\u7406\u6570 \\(c' , d'\\) \u3092\u6c42\u3081\u308b\u3068\r\n\\[\\begin{align}\r\n{\\alpha}^3 & = ( \\alpha +1 ) \\alpha = 2 \\alpha +1 = 2 +\\sqrt{5} , \\\\\r\n{\\beta}^3 & = ( \\beta +1 ) \\beta = 2 \\beta +1 = 2 -\\sqrt{5}\r\n\\end{align}\\]\r\n\\(x _ 2 = \\dfrac{1}{3}\\) \u3068, \u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nc' {\\alpha}^3 +d' {\\beta}^3 & = 2 ( c' +d' ) +\\sqrt{5} ( c' -d' ) = \\dfrac{1}{3}\r\n\\end{align}\\]\r\n\\(c' , d'\\) \u306f\u6709\u7406\u6570, \\(\\sqrt{5}\\) \u306f\u7121\u7406\u6570\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n2 ( c' +d' ) = \\dfrac{1}{3} & , \\ c' -d' = 0 \\\\\r\n\\text{\u2234} \\quad c' = d' & = \\dfrac{1}{12}\r\n\\end{align}\\]\r\n\u3086\u3048\u306b \\(c' = c\\) , \\(d' = d\\) \u306a\u306e\u3067, [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    3. 3*<\/strong>\u3000\\(n = k , k+1 \\ ( k = 1, 2, 3, \\cdots )\\) \u306e\u3068\u304d\u306b, [A] \u304c\u6210\u7acb\u3059\u308b, \u3059\u306a\u308f\u3061\r\n\\[\\begin{align}\r\nx _ k & = c {\\alpha}^{k+1} +d {\\beta}^{k+1} , \\\\\r\nx _ {k+1} & = c {\\alpha}^{k+2} +d {\\beta}^{k+2}\r\n\\end{align}\\]\r\n\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nx _ {k+2} & = x _ {k+1} +x _ k \\\\\r\n& = c {\\alpha}^{k+1} ( \\alpha +1 ) +d {\\beta}^{k+1} ( \\beta +1 ) \\\\\r\n& = c {\\alpha}^{k+2} +d {\\beta}^{k+2}\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \\(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, \u6c42\u3081\u308b\u4e00\u822c\u9805\u306f\r\n\\[\r\nr _ n = \\underline{\\dfrac{144}{\\left( {\\alpha}^{n+1} +{\\beta}^{n+1} \\right)^2}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5e73\u9762\u4e0a\u306e\u76f4\u7dda \\(\\ell\\) \u306b\u540c\u3058\u5074\u3067\u63a5\u3059\u308b \\(2\\) \u3064\u306e\u5186 \\(C _ 1 , C _ 2\\) \u304c\u3042\u308a, \\(C _ 1\\) \u3068 \\(C _ 2\\) \u3082\u4e92\u3044\u306b\u5916\u63a5\u3057\u3066\u3044\u308b. \\(\\ell , C _ 1 , 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":[121],"tags":[144,112],"class_list":["post-1178","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2014","tag-tsukuba_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1178","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=1178"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1178\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1178"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1178"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}