\u30b5\u30a4\u30b3\u30ed\u3092 \\(n\\) \u56de\u6295\u3052, \\(k\\) \u56de\u76ee\u306b\u51fa\u305f\u76ee\u3092 \\(a _ k\\) \u3068\u3059\u308b.\r\n\u307e\u305f, \\(s _ n\\) \u3092 \\(s _ n = \\sum\\limits _ {k=1}^{n} 10^{n-k} a _ k\\) \u3067\u5b9a\u3081\u308b.<\/p>\r\n
(1)<\/strong>\u3000\\(s _ n\\) \u304c \\(4\\) \u3067\u5272\u308a\u5207\u308c\u308b\u78ba\u7387\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(s _ n\\) \u304c \\(6\\) \u3067\u5272\u308a\u5207\u308c\u308b\u78ba\u7387\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(s _ n\\) \u304c \\(7\\) \u3067\u5272\u308a\u5207\u308c\u308b\u78ba\u7387\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u6c42\u3081\u308b\u78ba\u7387\u3092 \\(p _ n\\) \u3068\u304a\u304f. 1*<\/strong>\u3000\\(n = 1\\) \u306e\u3068\u304d 2*<\/strong>\u3000\\(n \\geqq 2\\) \u306e\u3068\u304d \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u78ba\u7387\u306f\r\n\\[\r\np _ n = \\underline{\\left\\{ \\begin{array}{ll} \\dfrac{1}{6} & ( \\ n = 1 \\text{\u306e\u3068\u304d} ) \\\\ \\dfrac{1}{4} & ( \\ n \\geqq 2 \\text{\u306e\u3068\u304d} ) \\end{array} \\right.}\r\n\\]\r\n (2)<\/strong><\/p>\r\n \u6c42\u3081\u308b\u78ba\u7387\u3092 \\(q _ n\\) \u3068\u304a\u304f.<\/p>\r\n 1*<\/strong>\u3000\\(n = 1\\) \u306e\u3068\u304d 2*<\/strong>\u3000\\(n \\geqq 2\\) \u306e\u3068\u304d \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u78ba\u7387\u306f\r\n\\[\r\nq _ n = \\underline{\\dfrac{1}{6}}\r\n\\]\r\n (3)<\/strong><\/p>\r\n \u6c42\u3081\u308b\u78ba\u7387\u3092 \\(r _ n\\) \u3068\u304a\u304f.<\/p>\r\n 1*<\/strong>\u3000\\(n = 1\\) \u306e\u3068\u304d 2*<\/strong>\u3000\\(n \\geqq 2\\) \u306e\u3068\u304d \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u78ba\u7387\u306f\r\n\\[\r\nr _ n = \\underline{\\dfrac{1}{7} \\left\\{ 1 -\\left( \\dfrac{1}{6} \\right)^{n-1} \\right\\}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u30b5\u30a4\u30b3\u30ed\u3092 \\(n\\) \u56de\u6295\u3052, \\(k\\) \u56de\u76ee\u306b\u51fa\u305f\u76ee\u3092 \\(a _ k\\) \u3068\u3059\u308b. \u307e\u305f, \\(s _ n\\) \u3092 \\(s _ n = \\sum\\limits _ {k=1}^{n} 10^{n-k} a _ k … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n\\(100 = 4 \\cdot 25\\) \u306a\u306e\u3067, \\(s _ n\\) \u306e\u4e0b \\(2\\) \u6841\u306b\u7740\u76ee\u3059\u308c\u3070\u3088\u3044.<\/p>\r\n\r\n
\r\n\\(4\\) \u304c\u51fa\u305f\u3068\u304d\u6761\u4ef6\u3092\u307f\u305f\u3059\u306e\u3067\r\n\\[\r\np _ 1 = \\dfrac{1}{6}\r\n\\]<\/li>\r\n
\r\n\u4e0b \\(2\\) \u6841\u304c, \\(12 , 16 , 24 , 32 , 36 , 44 , 52 , 56, 64\\) \u306e \\(9\\) \u901a\u308a\u304c\u6761\u4ef6\u3092\u307f\u305f\u3059\u306e\u3067\r\n\\[\r\np _ n = \\dfrac{9}{6^2} = \\dfrac{1}{4}\r\n\\]<\/li>\r\n<\/ol>\r\n\r\n
\r\n\\(6\\) \u304c\u51fa\u305f\u3068\u304d\u6761\u4ef6\u3092\u307f\u305f\u3059\u306e\u3067\r\n\\[\r\nq _ 1 = \\dfrac{1}{6}\r\n\\]<\/li>\r\n
\r\n\\(s _ {n+1} = 10 s _ n +a _ {n+1}\\) \u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3068, \\(6\\) \u3092\u6cd5\u3068\u3057\u3066 \\(s _ {n+1} \\equiv 0\\) \u3092\u3068\u304f\u3068\r\n\\[\\begin{align}\r\n10 s _ n + a _ {n+1} & \\equiv 0 \\\\\r\n\\text{\u2234} \\quad a _ {n+1} & \\equiv -4 s _ n \\equiv 2 s _ n\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(s _ n\\) \u306b\u5fdc\u3058\u3066, \u6761\u4ef6\u3092\u307f\u305f\u3059 \\(a _ {n+1}\\) \u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|cccccc} s _ n & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline a _ {n+1} & 6 & 2 & 4 & 6 & 2 & 4 \\end{array}\r\n\\]\r\n\u305d\u308c\u305e\u308c\u306e\u5834\u5408\u306b\u3064\u3044\u3066, \u6761\u4ef6\u3092\u307f\u305f\u3059\u76ee\u306f \\(1\\) \u3064\u305a\u3064\u3057\u304b\u306a\u3044\u306e\u3067\r\n\\[\r\nq _ n = \\dfrac{1}{6}\r\n\\]<\/li>\r\n<\/ol>\r\n\r\n
\r\n\u3069\u306e\u76ee\u304c\u51fa\u3066\u3082, \u6761\u4ef6\u3092\u307f\u305f\u3055\u306a\u3044\u306e\u3067\r\n\\[\r\nr _ 1 = 0 \\quad ... [1]\r\n\\]<\/li>\r\n
\r\n\\(7\\) \u3092\u6cd5\u3068\u3057\u3066, (2)<\/strong> \u3068\u540c\u69d8\u306b\u8003\u3048\u308b.
\r\n\\(s _ {n+1} \\equiv 0\\) \u3092\u3068\u304f\u3068\r\n\\[\\begin{align}\r\n10 s _ n + a _ {n+1} & \\equiv 0 \\\\\r\n\\text{\u2234} \\quad a _ {n+1} \\equiv -3 s _ n & \\equiv 4 s _ n\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(s _ n\\) \u306b\u5fdc\u3058\u3066, \u6761\u4ef6\u3092\u307f\u305f\u3059 \\(a _ {n+1}\\) \u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccccc} s _ n & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline a _ {n+1} & \\times & 4 & 1 & 5 & 2 & 6 & 3 \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nr _ {n+1} & = 0 \\cdot r _ n +\\dfrac{1}{6} ( 1-r _ n )\r\n\\\\\r\nr _ {n+1} -\\dfrac{1}{7} & = -\\dfrac{1}{6} \\left( r _ n -\\dfrac{1}{7} \\right)\r\n\\end{align}\\]\r\n[1] \u3082\u7528\u3044\u308c\u3070, \u6570\u5217 \\(\\left\\{ r _ n -\\dfrac{1}{7} \\right\\}\\) \u306f, \u521d\u9805 \\(r _ 1 -\\dfrac{1}{7} = -\\dfrac{1}{7}\\) , \u516c\u6bd4 \\(-\\dfrac{1}{6}\\) \u306e\u7b49\u6bd4\u6570\u5217\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nr _ n -\\dfrac{1}{7} & = -\\dfrac{1}{7} \\left( \\dfrac{1}{6} \\right)^{n-1}\r\n\\\\\r\n\\text{\u2234} \\quad r _ n & = \\dfrac{1}{7} \\left\\{ 1 -\\left( \\dfrac{1}{6} \\right)^{n-1} \\right\\}\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n