\u8907\u7d20\u6570 \\(z = 1+2 \\sqrt{6} i\\) \u3068\u81ea\u7136\u6570 \\(n = 1, 2, 3, \\cdots\\) \u306b\u3064\u3044\u3066, \u8907\u7d20\u6570 \\(z^n\\) \u3092\u5b9f\u6570 \\(a _ n , b _ n\\) \u3092\u7528\u3044\u3066\r\n\\[\r\nz^n = a _ n +b _ n i\r\n\\]\r\n\u3068\u8868\u3059. \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\\({a _ n}^2 +{b _ n}^2 = 5^{2n} \\ ( n = 1, 2, 3, \\cdots )\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\u3059\u3079\u3066\u306e \\(n\\) \u306b\u3064\u3044\u3066 \\(a _ {n+2} = pa _ {n+1} +qa _ {n}\\) \u304c\u6210\u308a\u7acb\u3064\u5b9a\u6570 \\(p , q\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (3)<\/strong>\u3000\u3069\u3093\u306a \\(n\\) \u306b\u3064\u3044\u3066\u3082 \\(a _ n\\) \u306f \\(5\\) \u306e\u6574\u6570\u500d\u3067\u306f\u306a\u3044\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (4)<\/strong>\u3000\\(z^n \\ ( n = 1, 2, 3, \\cdots )\\) \u306f\u5b9f\u6570\u3067\u306a\u3044\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u6761\u4ef6\u3088\u308a\r\n\\[\r\n|z| = \\sqrt{1^2 +\\left( 2 \\sqrt{6} \\right)^2} = 5\r\n\\]\r\n\u4e00\u822c\u306b\u8907\u7d20\u6570 \\(z\\) \u306b\u5bfe\u3057\u3066, \\(\\left| z^n \\right| = |z|^n\\) \u306a\u306e\u3067\r\n\\[\r\n{a _ n}^2 +{b _ n}^2 = \\left| z^n \\right|^2 = |z|^{2n} = 5^{2n}\r\n\\]\r\n (2)<\/strong><\/p>\r\n \\[\\begin{align}\r\nz^{n+1} & = \\left( 1 +2 \\sqrt{6} i \\right) \\left( a _ n +b _ n i \\right) \\\\\r\n& = \\left( a _ n -2 \\sqrt{6} b _ n \\right) +\\left( 2 \\sqrt{6} a _ n +b _ n \\right) i\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\left\\{\\begin{array}{ll} a _ {n+1} = a _ n -2 \\sqrt{6} b _ n & \\ ... [1] \\\\ b _ {n+1} = 2 \\sqrt{6} a _ n +b _ n & \\ ... [2] \\end{array}\\right.\r\n\\]\r\n[1] \u3088\u308a\r\n\\[\r\n2 \\sqrt{6} b _ n = a _ {n+1} -a _ n\r\n\\]\r\n\u3053\u308c\u3092 \\([2] \\times 2 \\sqrt{6}\\) \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\na _ {n+1} -a _ {n+2} & = 24a _ n +a _ n -a _ {n+1} \\\\\r\n\\text{\u2234} \\quad a _ {n+2} & = 2 a _ {n+1} -25a _ n \\quad ... [3]\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\np = \\underline{2} , \\ q = \\underline{-25}\r\n\\]\r\n (3)<\/strong><\/p>\r\n \u304c, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n 1*<\/strong>\u3000\\(n = 1\\) \u306e\u3068\u304d 2*<\/strong>\u3000\\(n = 2\\) \u306e\u3068\u304d 3*<\/strong>\u3000\\(n = k , k+1\\) \u306e\u3068\u304d\u306b, [A] \u304c\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3059\u308b. \u4ee5\u4e0a\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n (4)<\/strong><\/p>\r\n \\(b _ n \\neq 0\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n
\r\n
\r\n\\(a _ 1 = 1\\) \u306a\u306e\u3067, [A] \u306f\u6210\u308a\u7acb\u3064.<\/p><\/li>\r\n
\r\n[1] \u3088\u308a\r\n\\[\r\na _ 2 = 1 -2 \\sqrt{6} \\cdot 2 \\sqrt{6} = -23\r\n\\]\r\n\u306a\u306e\u3067, [A] \u306f\u6210\u308a\u7acb\u3064.<\/p><\/li>\r\n
\r\n\u3053\u3053\u3067, \\(a _ {k+2}\\) \u304c \\(5\\) \u306e\u500d\u6570\u3067\u3042\u308b, \u3059\u306a\u308f\u3061\r\n\\[\r\na _ {n+2} = 5M \\quad ( \\ M \\text{\u306f\u6574\u6570} )\r\n\\]\r\n\u3068\u4eee\u5b9a\u3059\u308b\u3068, [3] \u3088\u308a\r\n\\[\r\n2 a _ {k+1} = -a _ {k+2} -25a _ k = -5 ( M +5a _ k )\r\n\\]\r\n\u306a\u306e\u3067, \\(a _ {k+1}\\) \u306f \\(5\\) \u306e\u500d\u6570\u3068\u306a\u308b\u304c, \u3053\u308c\u306f\u77db\u76fe\u3067\u3042\u308b.
\r\n\u3057\u305f\u304c\u3063\u3066, \\(a _ {k+2}\\) \u306f \\(5\\) \u306e\u500d\u6570\u3067\u306f\u306a\u304f, \\(n = k+2\\) \u306e\u3068\u304d [A] \u304c\u6210\u308a\u7acb\u3064.<\/p><\/li>\r\n<\/ol>\r\n
\r\n\\(b _ n\\) \u304c \\(5\\) \u306e\u500d\u6570, \u3059\u306a\u308f\u3061\r\n\\[\r\nb _ n = 5N \\quad ( \\ N \\text{\u306f\u6574\u6570} )\r\n\\]\r\n\u3068\u4eee\u5b9a\u3059\u308b\u3068, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n{a _ n}^2 = 5^{2n} -(5N)^2 = 5^2 \\left( 5^{2(n-1)} -N \\right)\r\n\\]\r\n\u306a\u306e\u3067, \\(a _ n\\) \u306f \\(5\\) \u306e\u500d\u6570\u3068\u306a\u308b\u304c, \u3053\u308c\u306f (3)<\/strong> \u306e\u7d50\u679c\u306b\u77db\u76fe\u3059\u308b.
\r\n\u3088\u3063\u3066, \\(b _ n\\) \u306f \\(5\\) \u306e\u500d\u6570\u3067\u306f\u306a\u3044\u306e\u3067\r\n\\[\r\nb _ n \\neq 0\r\n\\]\r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u8907\u7d20\u6570 \\(z = 1+2 \\sqrt{6} i\\) \u3068\u81ea\u7136\u6570 \\(n = 1, 2, 3, \\cdots\\) \u306b\u3064\u3044\u3066, \u8907\u7d20\u6570 \\(z^n\\) \u3092\u5b9f\u6570 \\(a _ n , b _ n\\) \u3092\u7528\u3044\u3066 \\[ z^n = … \u7d9a\u304d\u3092\u8aad\u3080