{"id":795,"date":"2013-05-17T01:14:30","date_gmt":"2013-05-16T16:14:30","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=795"},"modified":"2021-03-24T21:36:42","modified_gmt":"2021-03-24T12:36:42","slug":"tok201301","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201301\/","title":{"rendered":"\u6771\u5de5\u59272013\uff1a\u7b2c1\u554f"},"content":{"rendered":"<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(2\\) \u6b21\u65b9\u7a0b\u5f0f \\(x^2-3x+5 = 0\\) \u306e \\(2\\) \u3064\u306e\u89e3 \\(\\alpha , \\beta\\) \u306b\u5bfe\u3057,\r\n\\({\\alpha}^n +{\\beta}^n -3^n\\) \u306f\u3059\u3079\u3066\u306e\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u3064\u3044\u3066 \\(5\\) \u306e\u6574\u6570\u500d\u306b\u306a\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(6\\) \u500b\u306e\u3055\u3044\u3053\u308d\u3092\u540c\u6642\u306b\u6295\u3052\u308b\u3068\u304d, \u3061\u3087\u3046\u3069 \\(4\\) \u7a2e\u985e\u306e\u76ee\u304c\u51fa\u308b\u78ba\u7387\u3092\u65e2\u7d04\u5206\u6570\u3067\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n<hr>\r\n<!--more-->\r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\n\\alpha +\\beta = 3 , \\ \\alpha \\beta = 5\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u3066, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066<\/p>\r\n<ol>\r\n<li>[P] ... \u300c \\({\\alpha}^n +{\\beta}^2 -3^n\\) \u304c \\(5\\) \u306e\u6574\u6570\u500d\u3067\u3042\u308b. \u300d<\/li>\r\n<\/ol>\r\n<p>\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3067\u793a\u3059.<\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(n=1\\) \u306e\u3068\u304d\r\n\\[\r\n\\alpha +\\beta -3 = 3-3 = 0\r\n\\]\r\n\u3086\u3048\u306b, \\(n=1\\) \u306e\u3068\u304d [P] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(n=2\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n{\\alpha}^2 +{\\beta}^2 -3^2 & = ( \\alpha +\\beta )^2 -2 \\alpha \\beta -9 \\\\\r\n& = 3^2 -2 \\cdot 5 -9 = -10\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \\(n=2\\) \u306e\u3068\u304d [P] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<li><p><strong>3*<\/strong>\u3000\\(n = k , k+1 \\ ( k \\geqq 1 )\\) \u306e\u3068\u304d\u306b [P] \u304c\u6210\u7acb\u3059\u308b, \u3059\u306a\u308f\u3061, \u6574\u6570 \\(L , M\\) \u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\n{\\alpha}^k +{\\beta}^k -3^k & = 5L \\\\\r\n{\\alpha}^{k+1} +{\\beta}^{k+1} -3^{k+1} & = 5M\r\n\\end{align}\\]\r\n\u3068\u8868\u305b\u308b\u3068\u4eee\u5b9a\u3059\u308b.<br \/>\r\n\u3053\u306e\u3068\u304d\r\n\\[\\begin{align}\r\n{\\alpha}^{k+2} & +{\\beta}^{k+2} -3^{k+2} \\\\\r\n& = ( \\alpha +\\beta ) \\left( {\\alpha}^{k+1} +{\\beta}^{k+1} \\right) -\\alpha \\beta \\left( {\\alpha}^k +{\\beta}^k \\right) -3^{k+2} \\\\\r\n& = 3 \\left( 5M +3^{k+1} \\right) -5 \\left( 5L +3^k \\right) -3^{k+2} \\\\\r\n& = 5 \\left( 3M -5L -3^k \\right)\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \\(n = k+2\\) \u306e\u3068\u304d\u3082 [P] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n<p>\u4ee5\u4e0a\u3088\u308a, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066, [P] \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f.<\/p>\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\u3055\u3044\u3053\u308d\u306e\u76ee\u306e\u51fa\u65b9\u306f\u5168\u90e8\u3067 \\(6^6\\) \u901a\u308a\u3042\u308b.<br \/>\r\n\\(4\\) \u7a2e\u985e\u306e\u76ee\u3092 A, B, C, D \u3068\u304a\u304f\u3068, \\(6\\) \u500b\u306e\u3055\u3044\u3053\u308d\u306e\u76ee\u304c<\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000A \u304c \\(3\\) \u500b, B, C, D \u304c \\(1\\) \u500b\u305a\u3064<\/p><\/li>\r\n<li><p><strong>2*<\/strong>\u3000A, B \u304c \\(2\\) \u500b\u305a\u3064, C, D \u304c \\(1\\) \u500b\u305a\u3064<\/p><\/li>\r\n<\/ol>\r\n<p>\u306e \\(2\\) \u901a\u308a\u306e\u5834\u5408\u304c\u8003\u3048\u3089\u308c\u308b.<br \/>\r\n\u305d\u308c\u305e\u308c\u304c\u8d77\u3053\u308b\u78ba\u7387\u3092\u6c42\u3081\u308b.<\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong> \u306e\u5834\u5408<br \/>\r\n\u300c \\(4\\) \u7a2e\u985e\u306e\u76ee\u306e\u9078\u3073\u65b9\u300d\u3068\u300c\u9078\u3093\u3060\u76ee\u306e \\(6\\) \u500b\u306e\u3055\u3044\u3053\u308d\u3078\u306e\u5272\u5f53\u3066\u65b9\u300d\u306b\u5206\u3051\u3066\u8003\u3048\u308b\u3068\r\n<ul>\r\n<li>A \u306e\u9078\u3073\u65b9\u304c \\(6\\) \u901a\u308a, B \uff5e D \uff08\u533a\u5225\u3057\u306a\u3044\uff09\u306e\u9078\u3073\u65b9\u304c \\({} _ 5 \\text{C} {} _ 3 = 10\\) \u901a\u308a<\/li>\r\n<li>A \u304c\u51fa\u308b\u3055\u3044\u3053\u308d\u306e\u5272\u5f53\u3066\u65b9\u304c \\({} _ 6 \\text{C} {} _ 3= 20\\) \u901a\u308a, B \uff5e D \u306e\u5272\u5f53\u3066\u65b9\u304c \\(3! = 6\\) \u901a\u308a<\/li>\r\n<\/ul>\r\n\u3057\u305f\u304c\u3063\u3066, <strong>1*<\/strong> \u304c\u751f\u3058\u308b\u78ba\u7387\u306f\r\n\\[\r\n\\dfrac{6 \\cdot 10 \\cdot 20 \\cdot 6}{6^6} = \\dfrac{200}{6^4}\r\n\\]<\/li>\r\n<li><p><strong>2*<\/strong> \u306e\u5834\u5408<br \/>\r\n<strong>1*<\/strong> \u3068\u540c\u69d8\u306b\u8003\u3048\u308b\u3068\r\n<ul>\r\n<li>A, B \uff08\u533a\u5225\u3057\u306a\u3044\uff09\u306e\u9078\u3073\u65b9\u304c \\({} _ 6 \\text{C} {} _ 2 = 15\\) \u901a\u308a, C, D \uff08\u533a\u5225\u3057\u306a\u3044\uff09\u306e\u9078\u3073\u65b9\u304c \\({} _ 4 \\text{C} {} _ 2 = 6\\) \u901a\u308a<\/li>\r\n<li>A \u304c\u51fa\u308b\u3055\u3044\u3053\u308d\u306e\u5272\u5f53\u3066\u65b9\u304c \\({} _ 6 \\text{C} {} _ 2= 15\\) \u901a\u308a, B \u304c\u51fa\u308b\u3055\u3044\u3053\u308d\u306e\u5272\u5f53\u3066\u65b9\u304c \\({} _ 4 \\text{C} {} _ 2= 6\\) \u901a\u308a, C, D \u306e\u5272\u5f53\u3066\u65b9\u304c \\(2\\) \u901a\u308a<\/li>\r\n<\/ul>\r\n\u3057\u305f\u304c\u3063\u3066, <strong>2*<\/strong> \u304c\u751f\u3058\u308b\u78ba\u7387\u306f\r\n\\[\r\n\\dfrac{15 \\cdot 6 \\cdot 15 \\cdot 6 \\cdot 2}{6^6} = \\dfrac{450}{6^4}\r\n\\]<\/li>\r\n<\/ol>\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u78ba\u7387\u306f\r\n\\[\r\n\\dfrac{200 +450}{6^4} = \\underline{\\dfrac{325}{648}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"(1)\u3000\\(2\\) \u6b21\u65b9\u7a0b\u5f0f \\(x^2-3x+5 = 0\\) \u306e \\(2\\) \u3064\u306e\u89e3 \\(\\alpha , \\beta\\) \u306b\u5bfe\u3057, \\({\\alpha}^n +{\\beta}^n -3^n\\) \u306f\u3059\u3079\u3066\u306e\u6b63\u306e\u6574\u6570 \\ &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/tok201301\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[92],"tags":[141,111],"class_list":["post-795","post","type-post","status-publish","format-standard","hentry","category-toko_2013","tag-toko","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/795","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=795"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/795\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=795"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=795"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=795"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}