{"id":209,"date":"2011-12-02T23:23:27","date_gmt":"2011-12-02T14:23:27","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=209"},"modified":"2021-09-14T12:56:50","modified_gmt":"2021-09-14T03:56:50","slug":"iks200902","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/iks200902\/","title":{"rendered":"\u533b\u79d1\u6b6f\u79d1\u59272009\uff1a\u7b2c2\u554f"},"content":{"rendered":"<hr \/>\n<p>\u6b63\u306e\u5b9f\u6570 \\(a , b , c\\) \u3092\u4fc2\u6570\u3068\u3059\u308b \\(2\\) \u6b21\u5f0f \\(f(x)=ax^2+bx+c\\) \u306b\u95a2\u3057\u3066, \u6b21\u306e\u6761\u4ef6 C \u3092\u8003\u3048\u308b.<\/p>\r\n<ol>\r\n<li>\u6761\u4ef6 C\uff1a \\(3\\) \u3067\u5272\u308a\u5207\u308c\u306a\u3044\u3059\u3079\u3066\u306e\u6574\u6570 \\(x\\) \u306b\u3064\u3044\u3066 \\(f(x)\\) \u304c\u6574\u6570\u306b\u306a\u308b.<\/li>\r\n<\/ol>\r\n<p>\u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(f(x)\\) \u304c\u6761\u4ef6 C \u3092\u6e80\u305f\u3059\u3068\u304d, \\(g(x) = f(x+3) -f(x)\\) \u306f\u4fc2\u6570\u304a\u3088\u3073\u5b9a\u6570\u9805\u304c\u6574\u6570\u3068\u306a\u308b \\(1\\) \u6b21\u5f0f\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\u6761\u4ef6 C \u3092\u6e80\u305f\u3059 \\(f(x)\\) \u306e\u3046\u3061, \\(f(1) = 1\\) \u3068\u306a\u308b\u3082\u306e\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\u4ee5\u4e0b\u306e\u6761\u4ef6 C' \u304c\u6761\u4ef6 C \u3068\u540c\u5024\u3068\u306a\u308b\u3088\u3046\u306a\u81ea\u7136\u6570\u306e\u7d44 \\((m _ 1, m _ 2, m _ 3)\\) \u306e\u3046\u3061, \\(m _ 1+m _ 2+m _ 3\\) \u304c\u6700\u5c0f\u3068\u306a\u308b\u3082\u306e\u3092\u6c42\u3081\u3088.<\/p>\r\n<ol>\r\n<li>\u6761\u4ef6C'\uff1a \\(m _ 1b , \\ m _ 2b , \\ m _ 3b , \\ a+b+c\\) \u304c\u3044\u305a\u308c\u3082\u6574\u6570\u3068\u306a\u308b.<\/li>\r\n<\/ol><\/li>\r\n<li><p><strong>(4)<\/strong>\u3000\\(n\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. \u6761\u4ef6 C \u3092\u6e80\u305f\u3059 \\(f(x)\\) \u306e\u3046\u3061, \\(f(1) = n\\) \u3068\u306a\u308b\u3082\u306e\u306e\u500b\u6570\u3092 \\(n\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\\[\\begin{align}\r\ng(x) & = f(x+3) -f(x) \\\\\r\n& = a(x+3)^2 +b(x+3) +c -ax^2 -bx -c \\\\\r\n& = 6ax +3(3a+b)\n\\end{align}\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\nh(x) & = f(x+1) -f(x) \\\\\r\n& =a(x+1)^2 +b(x+1) +c -ax^2 -bx -c \\\\\r\n& = 2ax +a+b\n\\end{align}\\]\r\n\u3068\u304a\u304f\u3068, \u6761\u4ef6 C \u3088\u308a, \\(k= 0, 1, 2, \\cdots\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\nh(3k+1) = 6ak +(3a+b)\n\\]\r\n\u304c\u6574\u6570\u3068\u306a\u308b\u306e\u3067, \\(6a , 3a+b\\) \u306f\u3068\u3082\u306b\u6574\u6570\u3068\u306a\u308b.<br \/>\r\n\u3088\u3063\u3066, \\(3(3a+b)\\) \u3082\u6574\u6570\u3068\u306a\u308b\u306e\u3067, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n<p><strong>(2)<\/strong>\r\n\\[\r\nf(1) = a+b+c = 1 \\quad ... [1]\n\\]\r\n\u6761\u4ef6\u3088\u308a, \\(0 \\lt a \\lt 1 , \\ 0 \\lt b \\lt 1 , \\ 0 \\lt c \\lt 1 \\quad ... [2]\\) .\r\n<strong>(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(6a\\) \u304c\u6574\u6570\u3068\u306a\u308b\u306e\u3067, [2] \u3088\u308a \\(a = \\dfrac{\\ell}{6} \\ ( \\ell = 1, 2, \\cdots, 5 )\\) \u3068\u8868\u305b\u308b.<br \/>\r\n\u307e\u305f, \\(3a+b =A\\) \uff08 \\(A\\) \u306f\u6574\u6570\uff09\u3068\u304a\u3051\u3070\r\n\\[\r\nb = A-3b = A -\\dfrac{\\ell}{2}\n\\]\r\n\u3053\u308c\u304c [2] \u3092\u307f\u305f\u3059\u306e\u306f, \\(\\ell =1\\) \u306e\u3068\u304d\u306b\u9650\u3089\u308c\u308b.<br \/>\r\n\u3057\u305f\u304c\u3063\u3066, [1] \u3088\u308a\r\n\\[\r\nc = 1 -\\dfrac{1}{6} -\\dfrac{1}{2} = \\dfrac{1}{3}\n\\]\r\n\u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b \\(f(x)\\) \u306f\r\n\\[\r\nf(x) = \\underline{\\dfrac{x^2}{6} +\\dfrac{x}{2} +\\dfrac{1}{3}}\n\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n<p>\\(f(x) = 1\\) \u3068\u306a\u308b \\(f(x)\\) \u306b\u3064\u3044\u3066, \\((m _ 1, m _ 2, m _ 3)\\) \u306e\u7d44\u3092\u8003\u3048\u308b.<br \/>\r\n\u3053\u306e\u3068\u304d, [1] \u3088\u308a, \\(a+b+c\\) \u306f\u6574\u6570\u3067\u3042\u308b.<br \/>\r\n\u307e\u305f\r\n\\[\r\nm _ 1 b = \\dfrac{m _ 1}{2}\n\\]\r\n\u3053\u308c\u304c\u6574\u6570\u3068\u306a\u308b\u6700\u5c0f\u306e \\(m _ 1\\) \u306f, \\(m _ 1 =2\\) .<br \/>\r\n\u3055\u3089\u306b\r\n\\[\r\nm _ 2a+m _ 3b =\\dfrac{m _ 2 +3m _ 3}{6}\n\\]\r\n\u3053\u308c\u304c\u6574\u6570\u3068\u306a\u308b\u5834\u5408\u3092 \\(m _ 2 +m _ 3\\) \u304c\u5c0f\u3055\u3044\u9806\u306b\u63a2\u3057\u3066\u3044\u304f\u3068<\/p>\r\n<ul>\r\n<li><p>\\(m _ 2+m _ 3=2\\) \u306e\u3068\u304d<br \/>\r\n\\(( m _ 2 , m _ 3 ) = ( 1, 1 )\\) \u304c\u5019\u88dc\u3067\u3042\u308b\u304c, \u6574\u6570\u3068\u306a\u3089\u305a\u4e0d\u9069.<\/p><\/li>\r\n<li><p>\\(m _ 2+m _ 3=3\\) \u306e\u3068\u304d<br \/>\r\n\\(( m _ 2 , m _ 3 ) = ( 2, 1 ) , ( 1, 2 )\\) \u304c\u5019\u88dc\u3067\u3042\u308b\u304c, \u3044\u305a\u308c\u3082\u6574\u6570\u3068\u306a\u3089\u305a\u4e0d\u9069.<\/p><\/li>\r\n<li><p>\\(m _ 2+m _ 3=4\\) \u306e\u3068\u304d<br \/>\r\n\\(( m _ 2 , m _ 3 ) = ( 3, 1 ) , ( 2, 2 ) , ( 1, 3 )\\) \u304c\u5019\u88dc\u3067\u3042\u308b.<br \/>\r\n\u3053\u306e\u3046\u3061, \\(( m _ 2 , m _ 3 ) = (3 , 1)\\) \u306e\u307f\u6574\u6570\u3068\u306a\u308b.<\/p><\/li>\r\n<\/ul>\r\n<p>\u4ee5\u4e0a\u3088\u308a, \u6761\u4ef6 C' \u306b\u304a\u3051\u308b\u5019\u88dc\u306f\r\n\\[\r\n( m _ 1, m _ 2, m _ 3 ) =( 2 , 3, 1 )\n\\]\r\n\u9006\u306b\u3053\u306e\u3068\u304d, \u6761\u4ef6 C \u304c\u6210\u7acb\u3059\u308b\u304b\u3092\u78ba\u8a8d\u3059\u308b.<br \/>\r\n\\(p =2b\\) , \\(q=3a+b\\) , \\(r=a+b+c\\) \uff08 \\(p , q , r\\) \u306f\u81ea\u7136\u6570\uff09...[3] \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf(1) & = a+b+c =r , \\\\\r\nf(2) & = 4a+2b+c = (a+b+c) +(3a+b) = q+r , \\\\\r\nf(x+3) -f(x) & = g(x) \\\\\r\n& = \\left\\{ 2(3a+b) -2b \\right\\}x +3(3a+b) = (2q-p)x +3q\n\\end{align}\\]\r\n\u3053\u308c\u3089\u306f\u3059\u3079\u3066\u6574\u6570\u3068\u306a\u308a, \u5e30\u7d0d\u7684\u306b\u6761\u4ef6 C \u304c\u6210\u7acb\u3059\u308b.<br \/>\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b \\(( m _ 1, m _ 2, m _ 3 )\\) \u306e\u7d44\u306f\r\n\\[\r\n( m _ 1, m _ 2, m _ 3 ) = \\underline{( 2 , 3 , 1 )}\n\\]\r\n<p><strong>(4)<\/strong><\/p>\r\n<p><strong>(3)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \u6761\u4ef6 C' \u3092\u6e80\u305f\u3059 \\(f(x)\\) \u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070\u3088\u3044.<br \/>\r\n\u6761\u4ef6\u3088\u308a\r\n\\[\r\nf(1) = a+b+c =r = n\n\\]\r\n[3] \u3068, \\(0 \\lt a \\lt n , \\ 0 \\lt b \\lt n , \\ 0 \\lt c \\lt n\\) \u3067\u3042\u308b\u3053\u3068\u304b\u3089<\/p>\r\n<ul>\r\n<li><p>\\(0 \\lt a+b \\lt n\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n0 & \\lt 3(a+b) = p+q \\lt 3n \\\\\r\n\\text{\u2234} \\quad 0 & \\lt q \\lt -p+3n \\quad ... [4]\n\\end{align}\\]<\/li>\r\n<li><p>\\(2b = p\\) \u306a\u306e\u3067\r\n\\[\r\n0 \\lt p \\lt 2n \\quad ... [5]\n\\]<\/li>\r\n<li><p>\\(3a = q-b = q -\\dfrac{p}{2}\\) \u306a\u306e\u3067\r\n\\[\r\n\\dfrac{p}{2} \\lt q \\lt \\dfrac{p}{2} +3n \\quad ... [6]\n\\]<\/li>\r\n<\/ul>\r\n[3] \u3088\u308a, \\(( a, b, c )\\) \u306e\u7d44\u3068 \\(( p, q, r )\\) \u306e\u7d44\u306f \\(1\\) \u5bfe \\(1\\) \u3067\u5bfe\u5fdc\u3057\u3066\u3044\u308b\u306e\u3067, [4] \uff5e [6] \u3092\u6e80\u305f\u3059\u6574\u6570 \\(( p, q )\\) \u306e\u7d44\u306e\u500b\u6570\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.<br \/>\r\n\u3053\u306e\u9818\u57df\u3092\u56f3\u793a\u3059\u308b\u3068\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u306f\u542b\u307e\u306a\u3044\uff09\u3068\u306a\u308b.<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/ikashika_200902_01.png\" alt=\"\" title=\"ikashika_200902_01\" class=\"aligncenter size-full\" \/>\r\n<p>\u3053\u306e\u9818\u57df\u306b\u542b\u307e\u308c, \u304b\u3064 \\(q = k \\ ( k= 1, 2, \\cdots , 3n-1 )\\) \u4e0a\u306b\u3042\u308b\u683c\u5b50\u70b9\u306e\u500b\u6570\u3092\u6570\u3048\u308c\u3070\u3088\u3044.<br \/>\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u500b\u6570\u306f\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {k=1}^{n-1} (2k-1) & +\\textstyle\\sum\\limits _ {k=n}^{3n-1} (3n -k-1) \\\\\r\n& = 2 \\cdot \\dfrac{n(n-1)}{2} -(n-1) +\\textstyle\\sum\\limits _ {k=1}^{2n} (k-1) \\\\\r\n& = (n-1)^2 +\\dfrac{2n(2n-1)}{2} \\\\\r\n& = \\underline{3n^2-3n+1}\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b63\u306e\u5b9f\u6570 \\(a , b , c\\) \u3092\u4fc2\u6570\u3068\u3059\u308b \\(2\\) \u6b21\u5f0f \\(f(x)=ax^2+bx+c\\) \u306b\u95a2\u3057\u3066, \u6b21\u306e\u6761\u4ef6 C \u3092\u8003\u3048\u308b. \u6761\u4ef6 C\uff1a \\(3\\) \u3067\u5272\u308a\u5207\u308c\u306a\u3044\u3059\u3079\u3066\u306e\u6574\u6570 \\(x\\) \u306b\u3064\u3044\u3066  &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/iks200902\/\">\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":[27],"tags":[145,15],"class_list":["post-209","post","type-post","status-publish","format-standard","hentry","category-ikashika_2009","tag-ikashika","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/209","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=209"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/209\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=209"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=209"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=209"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}