{"id":43,"date":"2011-11-25T22:14:29","date_gmt":"2011-11-25T13:14:29","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=43"},"modified":"2021-09-14T09:53:15","modified_gmt":"2021-09-14T00:53:15","slug":"iks201101","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/iks201101\/","title":{"rendered":"\u533b\u79d1\u6b6f\u79d1\u59272011\uff1a\u7b2c1\u554f"},"content":{"rendered":"<hr \/>\n<p>\u3042\u308b\u786c\u8ca8\u3092\u6295\u3052\u305f\u3068\u304d, \u8868\u3068\u88cf\u304c\u305d\u308c\u305e\u308c\u78ba\u7387 \\(\\dfrac{1}{2}\\) \u3067\u51fa\u308b\u3068\u3059\u308b.\r\n\u3053\u306e\u786c\u8ca8\u3092\u6295\u3052\u308b\u64cd\u4f5c\u3092\u7e70\u308a\u8fd4\u3057\u884c\u3044, \\(3\\) \u56de\u7d9a\u3051\u3066\u8868\u304c\u51fa\u305f\u3068\u304d\u3053\u306e\u64cd\u4f5c\u3092\u7d42\u4e86\u3059\u308b. \u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057,<\/p>\r\n<ul>\r\n<li><p>\u64cd\u4f5c\u304c\u3061\u3087\u3046\u3069 \\(n\\) \u56de\u76ee\u3067\u7d42\u4e86\u3068\u306a\u308b\u78ba\u7387\u3092 \\(P _ n\\)<\/p><\/li>\r\n<li><p>\u64cd\u4f5c\u304c \\(n\\) \u56de\u4ee5\u4e0a\u7e70\u308a\u8fd4\u3055\u308c\u308b\u78ba\u7387\u3092 \\(Q _ n\\)<\/p><\/li>\r\n<\/ul>\r\n<p>\u3068\u3059\u308b. \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\\(P _ 3 , P _ 4 , P _ 5 , P _ 6 , P _ 7\\) \u3092\u305d\u308c\u305e\u308c\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(Q _ 6 , Q _ 7\\) \u3092\u305d\u308c\u305e\u308c\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\\(n \\geqq 5\\) \u306e\u3068\u304d, \\(Q _ n -Q _ {n-1}\\) \u3092 \\(Q _ {n-4}\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<li><p><strong>(4)<\/strong>\u3000\\(n \\geqq 4\\) \u306e\u3068\u304d, \\(Q _ n \\lt \\left( \\dfrac{3}{4} \\right)^{\\frac{n-3}{4}}\\) \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\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>\u8868, \u88cf\u304c\u51fa\u308b\u3053\u3068\u3092\u305d\u308c\u305e\u308c\u25cb, \u00d7\u3067\u8868\u3059\u3053\u3068\u306b\u3059\u308b.<br \/>\r\n\\(3\\) \u56de\u76ee\u3067\u7d42\u4e86\u3059\u308b\u306e\u306f, \\(1\\) \u56de\u76ee\u304b\u3089\u300c\u25cb\u25cb\u25cb\u300d\u3068\u306a\u308b\u3068\u304d\u306a\u306e\u3067\r\n\\[\r\nP _ 3 = \\left( \\dfrac{1}{2} \\right)^3 = \\underline{\\dfrac{1}{8}}\n\\]\r\n\\(4\\) \u56de\u76ee\u3067\u7d42\u4e86\u3059\u308b\u306e\u306f, \\(1\\) \u56de\u76ee\u304b\u3089\u300c\u00d7\u25cb\u25cb\u25cb\u300d\u3068\u306a\u308b\u3068\u304d\u306a\u306e\u3067\r\n\\[\r\nP _ 4 = \\dfrac{1}{2} \\left( \\dfrac{1}{2} \\right)^3 = \\underline{\\dfrac{1}{16}}\n\\]\r\n\\(5\\) \u56de\u76ee\u3067\u7d42\u4e86\u3059\u308b\u306e\u306f, \\(1\\) \u56de\u76ee\u306f\u95a2\u4fc2\u306a\u304f, \\(2\\) \u56de\u76ee\u304b\u3089\u300c\u00d7\u25cb\u25cb\u25cb\u300d\u3068\u306a\u308b\u3068\u304d\u306a\u306e\u3067\r\n\\[\r\nP _ 5 = 1 \\cdot \\dfrac{1}{2} \\left( \\dfrac{1}{2} \\right)^3 = \\underline{\\dfrac{1}{16}}\n\\]\r\n\\(6\\) \u56de\u76ee\u3067\u7d42\u4e86\u3059\u308b\u306e\u306f, \\(1 , 2\\) \u56de\u76ee\u306f\u95a2\u4fc2\u306a\u304f, \\(3\\) \u56de\u76ee\u304b\u3089\u300c\u00d7\u25cb\u25cb\u25cb\u300d\u3068\u306a\u308b\u3068\u304d\u306a\u306e\u3067\r\n\\[\r\nP _ 6 = 1 \\cdot 1 \\cdot \\dfrac{1}{2} \\left( \\dfrac{1}{2} \\right)^3 = \\underline{\\dfrac{1}{16}}\n\\]\r\n\\(7\\) \u56de\u76ee\u3067\u7d42\u4e86\u3059\u308b\u306e\u306f, \\(3\\) \u56de\u76ee\u307e\u3067\u306b\u7d42\u4e86\u305b\u305a, \\(4\\) \u56de\u76ee\u304b\u3089\u300c\u00d7\u25cb\u25cb\u25cb\u300d\u3068\u306a\u308b\u3068\u304d\u306a\u306e\u3067\r\n\\[\r\nP _ 7 = ( 1-P _ 3 ) \\cdot \\dfrac{1}{2} \\left( \\dfrac{1}{2} \\right)^3 = \\underline{\\dfrac{7}{128}}\n\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\\(n \\geqq 4\\) \u306b\u3064\u3044\u3066\r\n\\[\r\nQ _ n = 1 -\\textstyle\\sum\\limits _ {k=3}^{n-1} P _ k , \\ Q _ n = Q _ {n-1} -P _ {n-1} \\quad ... [1]\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u7528\u3044\u308c\u3070,\r\n\\[\\begin{align}\r\nQ _ 6 & = 1 -\\left( P _ 3 +P _ 4 +P _ 5 \\right) = 1 -\\left( \\dfrac{1}{8} +2 \\cdot \\dfrac{1}{16} \\right) = \\underline{\\dfrac{3}{4}} , \\\\\r\nQ _ 7 & = Q _ 6 -P _ 6 = \\dfrac{3}{4} -\\dfrac{1}{16} = \\underline{\\dfrac{11}{16}}\n\\end{align}\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n[1] \u3088\u308a, \\(Q _ n -Q _ {n-1} = -P _ {n-1}\\) .<br \/>\r\n\\(n-1\\) \u56de\u76ee\u3067\u7d42\u4e86\u3059\u308b\u306e\u306f, \\(n-5\\) \u56de\u307e\u3067\u306b\u7d42\u4e86\u305b\u305a, \\(n-4\\) \u56de\u76ee\u304b\u3089\u300c\u00d7\u25cb\u25cb\u25cb\u300d\u3068\u306a\u308b\u3068\u304d\u306a\u306e\u3067\r\n\\[\r\nQ _ n -Q _ {n-1} = -\\dfrac{1}{2} \\left( \\dfrac{1}{2} \\right)^3 Q _ {n-4} = \\underline{-\\dfrac{Q _ {n-4}}{16}}\n\\]\r\n<p><strong>(4)<\/strong><\/p>\r\n<p>\\[\r\nQ _ n \\lt \\left( \\dfrac{3}{4} \\right)^{\\frac{n-3}{4}} \\quad ... [\\text{A}]\r\n\\]\r\n\u304c \\(n \\geqq 4\\) \u306b\u3064\u3044\u3066\u6210\u7acb\u3059\u308b\u3053\u3068\u3092, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<br \/>\r\n\\(n = 4k+l \\ ( k \\geqq 1 , \\ l = 0, 1, 2, 3 )\\) \u3068\u8868\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(k = 1\\) \u306e\u3068\u304d<br \/>\r\n\\(\\dfrac{7^4 \\cdot 4}{8^4 \\cdot 3} = \\dfrac{1372 \\cdot 7}{1536 \\cdot 8} \\lt 1\\) \u306a\u306e\u3067\r\n\\[\r\nQ _ 4 = 1-\\dfrac{1}{8} =\\dfrac{7}{8} \\lt \\left( \\dfrac{3}{4} \\right)^{\\frac{1}{4}}\n\\]\r\n\\(\\dfrac{13^2 \\cdot 4}{16^2 \\cdot 3} = \\dfrac{676}{768} \\lt 1\\) \u306a\u306e\u3067\r\n\\[\r\nQ _ 5 =\\dfrac{7}{8} -\\dfrac{1}{16} =\\dfrac{13}{16} \\lt \\left( \\dfrac{3}{4} \\right)^{\\frac{1}{2}}\n\\]\r\n\u3055\u3089\u306b\r\n\\[\r\nQ _ 6 = \\dfrac{3}{4} \\lt \\left( \\dfrac{3}{4} \\right)^{\\frac{3}{4}} , \\ Q _ 7 = \\dfrac{11}{16} \\lt \\dfrac{3}{4}\n\\]\r\n\u306a\u306e\u3067, \\(k = 1\\) \u306e\u3068\u304d, [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(k = m \\ ( m \\geqq 1 )\\) \u306e\u3068\u304d<br \/>\r\n[A]\u304c\u6210\u7acb\u3057\u3066\u3044\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068, \\(l = 1\\) \u306e\u3068\u304d, <strong>(3)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\nQ _ {4(k+1)+l} & -\\left( \\dfrac{3}{4} \\right)^{\\frac{4(k+1)+l-3}{4}} \\\\\r\n& = Q _ {4(k+1)+l-1} -\\dfrac{1}{16} Q _ {4k+l} -\\left( \\dfrac{3}{4} \\right)^{\\frac{4k+l+1}{4}} \\\\\r\n& = \\left( \\dfrac{3}{4} \\right)^{\\frac{4k+l}{4}} -\\dfrac{1}{16} \\left( \\dfrac{3}{4} \\right)^{\\frac{4k+l-3}{4}} -\\left( \\dfrac{3}{4} \\right)^{\\frac{4k+l+1}{4}} \\\\\r\n& = \\left\\{ \\left( \\dfrac{3}{4} \\right)^{\\frac{3}{4}} -\\dfrac{1}{16} -\\dfrac{3}{4} \\right\\} \\left( \\dfrac{3}{4} \\right)^{\\frac{4k+l-3}{4}} \\\\\r\n& \\lt \\left( \\dfrac{3}{4} -\\dfrac{13}{16} \\right) \\left( \\dfrac{3}{4} \\right)^{\\frac{4k+l-3}{4}} \\lt 0 \\\\\r\n\\text{\u2234} \\quad & Q _ {4(k+1)+l} \\lt \\left( \\dfrac{3}{4} \\right)^{\\frac{4(k+1)+l-3}{4}} \\quad ... [2]\n\\end{align}\\]\r\n\u540c\u69d8\u306b\u3059\u308c\u3070, \\(l = 2, 3, 4\\) \u306b\u3064\u3044\u3066\u3082, \u9806\u6b21 [2] \u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067, \\(k = m+1\\) \u306e\u3068\u304d\u3082, [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n<p><strong>1*<\/strong> <strong>2*<\/strong> \u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u3042\u308b\u786c\u8ca8\u3092\u6295\u3052\u305f\u3068\u304d, \u8868\u3068\u88cf\u304c\u305d\u308c\u305e\u308c\u78ba\u7387 \\(\\dfrac{1}{2}\\) \u3067\u51fa\u308b\u3068\u3059\u308b. \u3053\u306e\u786c\u8ca8\u3092\u6295\u3052\u308b\u64cd\u4f5c\u3092\u7e70\u308a\u8fd4\u3057\u884c\u3044, \\(3\\) \u56de\u7d9a\u3051\u3066\u8868\u304c\u51fa\u305f\u3068\u304d\u3053\u306e\u64cd\u4f5c\u3092\u7d42\u4e86\u3059\u308b. \u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057, \u64cd &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/iks201101\/\">\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":[29],"tags":[145,13],"class_list":["post-43","post","type-post","status-publish","format-standard","hentry","category-ikashika_2011","tag-ikashika","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/43","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=43"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/43\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=43"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=43"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=43"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}