{"id":1977,"date":"2021-11-18T08:44:36","date_gmt":"2021-11-17T23:44:36","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1977"},"modified":"2021-11-19T14:59:06","modified_gmt":"2021-11-19T05:59:06","slug":"tok202101","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok202101\/","title":{"rendered":"\u6771\u5de5\u59272021\uff1a\u7b2c1\u554f"},"content":{"rendered":"<hr \/>\n<p>\u6b63\u306e\u6574\u6570\u306b\u95a2\u3059\u308b\u6761\u4ef6<\/p>\r\n<ol>\r\n<li>(\uff0a)\u3000\\(10\\) \u9032\u6cd5\u3067\u8868\u3057\u305f\u3068\u304d\u306b, \u3069\u306e\u4f4d\u306b\u3082\u6570\u5b57 \\(9\\) \u304c\u73fe\u308c\u306a\u3044.<\/li>\r\n<\/ol>\r\n<p>\u3092\u8003\u3048\u308b. \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(k\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b\u3068\u304d, \r\n\\(10^{k-1}\\) \u4ee5\u4e0a\u304b\u3064 \\(10^k\\) \u672a\u6e80\u3067\u3042\u3063\u3066\u6761\u4ef6 (\uff0a) \u3092\u6e80\u305f\u3059\u6b63\u306e\u6574\u6570\u306e\u500b\u6570\u3092 \\(a_k\\) \u3068\u3059\u308b.\r\n\u3053\u306e\u3068\u304d, \\(a_k\\) \u3092 \\(k\\) \u306e\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066,\r\n\\[\r\nb_n = \\left\\{ \\begin{array}{ll} \\dfrac{1}{n} & ( \\ n \\ \\text{\u304c\u6761\u4ef6 (\uff0a) \u3092\u6e80\u305f\u3059\u3068\u304d} \\ ) \\\\ 0 & ( \\ n \\ \\text{\u304c\u6761\u4ef6 (\uff0a) \u3092\u6e80\u305f\u3055\u306a\u3044\u3068\u304d} \\ ) \\end{array} \\right.\r\n\\]\r\n\u3068\u304a\u304f.\r\n\u3053\u306e\u3068\u304d, \u3059\u3079\u3066\u306e\u6b63\u306e\u6574\u6570 \\(k\\) \u306b\u5bfe\u3057\u3066\u6b21\u306e\u4e0d\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.\r\n\\[\r\n\\textstyle\\sum\\limits _ {n=1}^{10^k -1} b_n \\lt 80\r\n\\]<\/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>\\(10^{k-1}\\) \u4ee5\u4e0a \\(10^k\\) \u672a\u6e80, \u3059\u306a\u308f\u3061 \\(k\\) \u6841\u306e\u6574\u6570\u306e\u3046\u3061, \u6761\u4ef6\u3092\u6e80\u305f\u3059\u3082\u306e\u306f<\/p>\r\n<ul>\r\n<li><p>\\(10^{k-1}\\) \u306e\u4f4d\u306e\u6570\u5b57\u304c \\(1\\) \uff5e \\(8\\)<\/p><\/li>\r\n<li><p>\u4ed6\u306e\u4f4d\u306e\u6570\u5b57\u304c \\(0\\) \uff5e \\(8\\)<\/p><\/li>\r\n<\/ul>\r\n<p>\u306a\u306e\u3067, \u6c42\u3081\u308b\u500b\u6570\u306f\r\n\\[\r\na_k = \\underline{8 \\cdot 9^{k-1}}\r\n\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\\(S _ m = \\textstyle\\sum\\limits _ {\\ell = 10^{m-1}}^{10^m -1} b_\\ell\\) \u3068\u304a\u304f.<br \/>\r\n\\(10^{m-1} \\leqq \\ell \\leqq 10^m -1\\) \u306b\u304a\u3044\u3066\r\n\\[\r\nb _ \\ell \\leqq \\dfrac{1}{10^{m-1}}\r\n\\]\r\n\u306a\u306e\u3067, <strong>(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nS _ m \\leqq \\dfrac{a _ m}{10^{m-1}} = 8 \\left( \\dfrac{9}{10} \\right)^{m-1}\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {n=1}^{10^k -1} b_n & = \\textstyle\\sum\\limits _ {m=1}^{k} S _ m \\\\\r\n& \\leqq 8 \\cdot \\dfrac{1 -\\left( \\dfrac{9}{10} \\right)^k}{1 -\\dfrac{9}{10}} \\\\\r\n& = 80 \\left\\{ 1 -\\left( \\dfrac{9}{10} \\right)^k \\right\\} \\lt 80\r\n\\end{align}\\]\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\n\\textstyle\\sum\\limits _ {n=1}^{10^k -1} b_n \\lt 80\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b63\u306e\u6574\u6570\u306b\u95a2\u3059\u308b\u6761\u4ef6 (\uff0a)\u3000\\(10\\) \u9032\u6cd5\u3067\u8868\u3057\u305f\u3068\u304d\u306b, \u3069\u306e\u4f4d\u306b\u3082\u6570\u5b57 \\(9\\) \u304c\u73fe\u308c\u306a\u3044. \u3092\u8003\u3048\u308b. \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\\(k\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b\u3068\u304d, \\(10^{k-1}\\) \u4ee5\u4e0a\u304b\u3064 &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/tok202101\/\">\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":[173],"tags":[141,165],"class_list":["post-1977","post","type-post","status-publish","format-standard","hentry","category-toko_2021","tag-toko","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1977","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=1977"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1977\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1977"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1977"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1977"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}