{"id":1437,"date":"2017-05-18T23:02:44","date_gmt":"2017-05-18T14:02:44","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1437"},"modified":"2021-09-08T14:45:43","modified_gmt":"2021-09-08T05:45:43","slug":"osr201601","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr201601\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2016\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(1\\) \u4ee5\u4e0a \\(6\\) \u4ee5\u4e0b\u306e \\(2\\) \u3064\u306e\u6574\u6570 \\(a , b\\) \u306b\u5bfe\u3057, \u95a2\u6570 \\(f _ n (x) = \\ ( n = 1, 2, 3, \\cdots )\\) \u3092\u6b21\u306e\u6761\u4ef6 (\u30a2), (\u30a4), (\u30a6) \u3067\u5b9a\u3081\u308b.\r\n\\[\r\n\\begin{array}{lll} \\text{(\u30a2)} & f _ 1 (x) = \\sin ( \\pi x ) & \\\\ \\text{(\u30a4)} & f _ {2n} (x) = f _ {2n-1} \\left( \\dfrac{1}{a} +\\dfrac{1}{b} -x \\right) & ( n = 1, 2, 3, \\cdots ) \\\\ \\text{(\u30a6)} & f _ {2n+1} (x) = f _ {2n} ( -x ) & ( n = 1, 2, 3, \\cdots ) \\end{array}\r\n\\]\r\n\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a = 2\\) , \\(b = 3\\) \u306e\u3068\u304d, \\(f _ 5 (0)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(a = 2\\) , \\(b = 3\\) \u306e\u3068\u304d, \\(\\textstyle\\sum\\limits _ {k=1}^{100} (-1)^k f _ {2k} (0)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(1\\) \u500b\u306e\u3055\u3044\u3053\u308d\u3092 \\(2\\) \u56de\u6295\u3052\u3066, \\(1\\) \u56de\u76ee\u306b\u51fa\u308b\u76ee\u3092 \\(a\\) , \\(2\\) \u56de\u76ee\u306b\u51fa\u308b\u76ee\u3092 \\(b\\) \u3068\u3059\u308b\u3068\u304d, \\(f _ 6 (0) = 0\\) \u3068\u306a\u308b\u78ba\u7387\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \\(c = \\dfrac{1}{a} +\\dfrac{1}{b}\\) \u3068\u304a\u304f.
    \r\n\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\nf _{2n} (x) = \\sin ( cn -x ) \\pi , \\ f _{2n+1} (x) = \\sin ( cn +x ) \\pi \\quad ... [ \\text{\uff0a} ]\r\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(n = 1\\) \u306e\u3068\u304d
      \r\n\u6761\u4ef6 (\u30a4) (\u30a6) \u3088\u308a\r\n\\[\\begin{align}\r\nf_2 (x) & = f_1 (c-x) = \\sin (c-x) \\pi \\\\\r\nf_3 (x) & = f_2 (-x) = \\sin (c+x) \\pi\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, [\uff0a] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(n = k\\) \u306e\u3068\u304d, [\uff0a] \u304c\u6210\u7acb\u3059\u308b, \u3059\u306a\u308f\u3061\r\n\\[\r\nf _{2k} (x) = \\sin ( ck -x ) \\pi , \\ f _{2k+1} (x) = \\sin ( ck +x ) \\pi\r\n\\]\r\n\u3068\u4eee\u5b9a\u3059\u308b\u3068, \u6761\u4ef6 (\u30a4) (\u30a6) \u3088\u308a\r\n\\[\\begin{align}\r\nf _{2k+2} (x) & = f _{2k+1} (c-x) = \\sin \\{ c(k+1) -x \\} \\pi \\\\\r\nf _{2k+3} (x) & = f _{2k+2} (-x) = \\sin \\{ c(k+1) +x \\} \\pi\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(n = k+1\\) \u306e\u3068\u304d\u3082 [\uff0a] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066, [\uff0a] \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f.<\/p>\r\n

      \\(c = \\dfrac{1}{2} +\\dfrac{1}{3} = \\dfrac{5}{6}\\) \u306a\u306e\u3067, [\uff0a] \u3088\u308a\r\n\\[\\begin{align}\r\nf_5 (0) & = \\sin 2 \\cdot \\dfrac{5}{6} \\pi \\\\\r\n& = \\sin \\dfrac{5 \\pi}{3} = \\underline{-\\dfrac{\\sqrt{3}}{2}}\r\n\\end{align}\\]\r\n

      (2)<\/strong><\/p>\r\n

      \\(g(n) = (-1)^n f _{2n} (0)\\) \u3068\u304a\u304f.
      \r\n\\(c = \\dfrac{5}{6}\\) \u306a\u306e\u3067\r\n\\[\r\ng(n) = (-1)^n \\sin \\dfrac{5 n \\pi}{6}\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\ng (n+6) & = (-1)^{n+6} f _{2(n+6)} (0) \\\\\r\n& = (-1)^n \\sin \\left( \\dfrac{5n}{6} +5 \\right) \\pi \\\\\r\n& = (-1)^{n+1} \\sin \\dfrac{5 n \\pi}{6} = -g(n)\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\ng (n+12) = g(n)\r\n\\]\r\n\u307e\u305f\r\n\\[\r\n\\textstyle\\sum\\limits _{k=i}^{i+12} g(k) = 0\r\n\\]\r\n\\(100 = 4 +8 \\cdot 12\\) \u306a\u306e\u3067, \u6c42\u3081\u308b\u548c\u306f\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _{k=1}^{100} g(k) & = \\textstyle\\sum\\limits _{k=1}^{4} g(k) \\\\\r\n& = \\sin \\dfrac{5 \\pi}{6} -\\sin \\dfrac{5 \\pi}{3} +\\sin \\dfrac{5 \\pi}{2} -\\sin \\dfrac{10 \\pi}{3} \\\\\r\n& = \\dfrac{1}{2} -\\left( -\\dfrac{\\sqrt{3}}{2} \\right) +1 -\\left( -\\dfrac{\\sqrt{3}}{2} \\right) \\\\\r\n& = \\underline{\\dfrac{3}{2} +\\sqrt{3}}\r\n\\end{align}\\]\r\n

      (3)<\/strong><\/p>\r\n[\uff0a] \u3088\u308a\r\n\\[\r\nf _6 (0) = \\sin 3c \\pi = 0\r\n\\]\r\n\u306a\u306e\u3067, \u6574\u6570 \\(m\\) \u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\n3c & = m \\\\\r\n3 \\left( \\dfrac{1}{a} +\\dfrac{1}{b} \\right) & = m \\\\\r\n\\text{\u2234} \\quad 3 (a+b) & = mab\r\n\\end{align}\\]\r\n\\(a , b\\) \u306f \\(1\\) \u304b\u3089 \\(6\\) \u306e\u6574\u6570\u306a\u306e\u3067, \u3053\u308c\u3092\u6e80\u305f\u3059 \\((a,b)\\) \u306e\u7d44\u306f\r\n\\[\r\n(1,1) , (1,3) , (2,2) , (2,6) , (3,1) , (3,3) , (6,2) , (6,6)\r\n\\]\r\n\u306e \\(8\\) \u7d44\u306e\u307f.
      \r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u78ba\u7387\u306f\r\n\\[\r\n\\dfrac{8}{6^2} = \\underline{\\dfrac{2}{9}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(1\\) \u4ee5\u4e0a \\(6\\) \u4ee5\u4e0b\u306e \\(2\\) \u3064\u306e\u6574\u6570 \\(a , b\\) \u306b\u5bfe\u3057, \u95a2\u6570 \\(f _ n (x) = \\ ( n = 1, 2, 3, \\cdots )\\) \u3092\u6b21\u306e\u6761\u4ef6 (\u30a2), (\u30a4), (\u30a6) … \u7d9a\u304d\u3092\u8aad\u3080 →<\/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":[152],"tags":[142,162],"class_list":["post-1437","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2016","tag-osaka_r","tag-162"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1437","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=1437"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1437\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1437"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1437"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1437"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}