{"id":126,"date":"2011-11-30T00:58:59","date_gmt":"2011-11-29T15:58:59","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=126"},"modified":"2021-09-24T18:04:19","modified_gmt":"2021-09-24T09:04:19","slug":"tok201002","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201002\/","title":{"rendered":"\u6771\u5de5\u59272010\uff1a\u7b2c2\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(a\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b. \u6b63\u306e\u5b9f\u6570 \\(x\\) \u306b\u3064\u3044\u3066\u306e\u65b9\u7a0b\u5f0f\r\n\\[\r\n\\text{(\uff0a)} \\quad x = \\left[ \\dfrac{1}{2} \\left( x + \\dfrac{a}{x} \\right) \\right]\r\n\\]\r\n\u304c\u89e3\u3092\u6301\u305f\u306a\u3044\u3088\u3046\u306a \\(a\\) \u3092\u5c0f\u3055\u3044\u9806\u306b\u4e26\u3079\u305f\u3082\u306e\u3092 \\(a _ 1 , a _ 2 , a _ 3 , \\cdots\\) \u3068\u3059\u308b.\r\n\u3053\u3053\u306b \\([ ]\\) \u306f\u30ac\u30a6\u30b9\u8a18\u53f7\u3067, \u5b9f\u6570 \\(u\\) \u306b\u5bfe\u3057, \\([ u ]\\) \u306f \\(u\\) \u4ee5\u4e0b\u306e\u6700\u5927\u306e\u6574\u6570\u3092\u8868\u3059.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(a = 7 , 8 , 9\\) \u306e\u5404\u3005\u306b\u3064\u3044\u3066 (\uff0a) \u306e\u89e3\u304c\u3042\u308b\u304b\u3069\u3046\u304b\u3092\u5224\u5b9a\u3057, \u3042\u308b\u5834\u5408\u306f\u89e3 \\(x\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(a _ 1 , a _ 2\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\\(\\textstyle\\sum\\limits _ {n=1}^{\\infty} \\dfrac{1}{a _ n}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p>(\uff0a) \u3088\u308a, \\(x\\) \u306f\u81ea\u7136\u6570\u3067\u3042\u308b. \u307e\u305f\r\n\\[\\begin{align}\r\nx & \\leqq \\dfrac{1}{2} \\left( x +\\dfrac{a}{x} \\right) \\lt x+1 \\\\\r\n2x^2 & \\leqq x^2 +a \\lt 2x^2+2x \\quad ( \\ \\text{\u2235} \\ x \\gt 0 ) \\\\\r\nx^2 -a & \\leqq 0 , \\ x^2 +2x -a \\gt 0 \\\\\r\n0 \\lt & x \\leqq \\sqrt{a} , \\ x \\gt -1+\\sqrt{a+1} \\quad ( \\ \\text{\u2235} \\ a \\gt 0 ) \\\\\r\n\\text{\u2234} \\quad & \\sqrt{a+1} -1 \\lt x \\leqq \\sqrt{a} \\quad ... [1]\r\n\\end{align}\\]\r\n\u3053\u3053\u3067 \\(k\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b\u3068,<\/p>\r\n<ul>\r\n<li><p>\\(k^2-1 \\leqq a \\lt ( k+1 )^2 -1\\) \u306e\u3068\u304d\r\n\\[\r\nk-1 \\leqq \\sqrt{a+1} -1 \\lt k\r\n\\]<\/li>\r\n<li><p>\\(k^2 \\leqq a \\lt ( k+1 )^2\\) \u306e\u3068\u304d\r\n\\[\r\nk \\leqq \\sqrt{a} \\lt k+1\r\n\\]<\/li>\r\n<\/ul>\r\n<p>\u3067\u3042\u308b\u3053\u3068\u304b\u3089<\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(a = k^2-1\\) \u306e\u3068\u304d<br \/>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/tokodai2010_02_01-300x104.png\" alt=\"\" title=\"tokodai2010_02_01\" class=\"aligncenter size-medium\" \/>\r\n\\(\\sqrt{a+1} -1 = k-1\\) , \\(\\sqrt{a} \\lt k\\) \u306a\u306e\u3067, [1] \u306f\u81ea\u7136\u6570\u89e3\u3092\u3082\u305f\u306a\u3044.<\/p><\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(k^2 \\leqq a \\lt ( k+1 )^2-1\\) \u306e\u3068\u304d<br \/>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/tokodai2010_02_02.png\" alt=\"\" title=\"tokodai2010_02_02\" class=\"aligncenter size-full\" \/>\r\n\\(\\sqrt{a+1} -1 \\lt k , \\ \\sqrt{a} \\geqq k\\) \u306a\u306e\u3067, [1] \u306e\u81ea\u7136\u6570\u89e3\u306f\r\n\\[\r\nx = k\r\n\\]<\/li>\r\n<\/ol>\r\n<p><strong>1*<\/strong> <strong>2*<\/strong>\u3088\u308a<\/p>\r\n<ul>\r\n<li><p>\\(a=7\\) \u306e\u3068\u304d\u306f, \\(\\underline{x=2}\\)<\/p><\/li>\r\n<li><p>\\(a=8\\) \u306e\u3068\u304d\u306f,<strong>\u89e3\u306a\u3057<\/strong><\/p><\/li>\r\n<li><p>\\(a=9\\) \u306e\u3068\u304d\u306f, \\(\\underline{x=3}\\)<\/p><\/li>\r\n<\/ul>\r\n<p><strong>(2)<\/strong><\/p>\r\n<p><strong>(1)<\/strong> \u3067\u306e\u8003\u5bdf\u3088\u308a, \\(a = k^2-1\\) \u3068\u8868\u3055\u308c\u308b\u3068\u304d, (\uff0a) \u306f\u89e3\u3092\u6301\u305f\u306a\u3044\u306e\u3067\r\n\\[\r\na _ n = (n+1)^2-1 =n(n+2) \\quad ... [2]\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\r\na _ 1 = 1 \\cdot 3 =\\underline{3} , \\ a _ 2 = 2 \\cdot 4 =\\underline{8}\r\n\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n[2] \u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {n=1}^{\\infty} \\dfrac{1}{a _ n} & = \\displaystyle\\lim _ {m \\rightarrow \\infty} \\textstyle\\sum\\limits _ {n=1}^{m} \\dfrac{1}{n(n+2)} \\\\\r\n& = \\displaystyle\\lim _ {m \\rightarrow \\infty} \\textstyle\\sum\\limits _ {n=1}^{m} \\dfrac{1}{2} \\left( \\dfrac{1}{n} -\\dfrac{1}{n+2} \\right) \\\\\r\n& = \\displaystyle\\lim _ {m \\rightarrow \\infty} \\dfrac{1}{2} \\left( 1 +\\dfrac{1}{2} -\\dfrac{1}{m+1} -\\dfrac{1}{m+2} \\right) \\\\\r\n& = \\underline{\\dfrac{3}{4}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b. \u6b63\u306e\u5b9f\u6570 \\(x\\) \u306b\u3064\u3044\u3066\u306e\u65b9\u7a0b\u5f0f \\[ \\text{(\uff0a)} \\quad x = \\left[ \\dfrac{1}{2} \\left( x + \\dfrac{a}{x} \\right &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/tok201002\/\">\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":[25],"tags":[141,14],"class_list":["post-126","post","type-post","status-publish","format-standard","hentry","category-toko_2010","tag-toko","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/126","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=126"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/126\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=126"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=126"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}