\u6570\u5217 \\(\\{ a _ n \\}, \\{ b _ n \\}, \\{ c _ n \\}\\) \u3092\r\n\\[\\begin{align}\r\na _ 1 & = 2 , \\ a _ {n+1} = 4 a _ n \\ . \\\\\r\nb _ 1 & = 3 , \\ b _ {n+1} = b _ n +2 a _ n \\ . \\\\\r\nc _ 1 & = 4 , \\ c _ {n+1} = \\dfrac{c _ n}{4} +a _ n +b _ n\r\n\\end{align}\\]\r\n\u3068\u9806\u306b\u5b9a\u3081\u308b. \u653e\u7269\u7dda \\(y = a _ n x^2 +2 b _ n x +c _ n\\) \u3092 \\(H _ n\\) \u3068\u3059\u308b.<\/p>\r\n
(1)<\/strong>\u3000\\(H _ n\\) \u306f \\(x\\) \u8ef8\u3068 \\(2\\) \u70b9\u3067\u4ea4\u308f\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(H _ n\\) \u3068 \\(x\\) \u8ef8\u3068\u306e\u4ea4\u70b9\u3092 \\(\\text{P}{} _ n , \\text{Q}{} _ n\\) \u3068\u3059\u308b. \\(\\textstyle\\sum\\limits _ {k=1}^n \\text{P}{} _ n \\text{Q}{} _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n 1*<\/strong>\u3000\\(n=1\\) \u306e\u3068\u304d\r\n\\[\r\n2x^2+6x+4 = 0\r\n\\]\r\n\u5224\u5225\u5f0f\u3092 \\(D _ 1\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\dfrac{D _ 1}{4} = 3^2 -2 \\cdot 4 = 1 \\gt 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(H _ 1\\) \u306f \\(x\\) \u8ef8\u3068 \\(2\\) \u70b9\u3067\u4ea4\u308f\u308b.<\/p><\/li>\r\n 2*<\/strong>\u3000\\(n = k \\ ( k \\geqq 1 )\\) \u306e\u3068\u304d \u4ee5\u4e0a\u3088\u308a, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066, \\(H _ n\\) \u306f \\(x\\) \u8ef8\u3068 \\(2\\) \u70b9\u3067\u4ea4\u308f\u308b.<\/p>\r\n (2)<\/strong><\/p>\r\n \\(a _ n x^2 +2 b _ n x +c _ n = 0\\) \u306e \\(2\\) \u89e3\u3092 \\({\\alpha} _ n , {\\beta} _ n \\ ( {\\alpha} _ n \\lt {\\beta} _ n )\\) \u3068\u304a\u304f\u3068, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\n{\\alpha} _ n + {\\beta} _ n = -\\dfrac{2 b _ n}{a _ n} , \\ {\\alpha} _ n {\\beta} _ n = \\dfrac{c _ n}{a _ n}\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\text{P}{} _ n \\text{Q}{} _ n & = {\\beta} _ n - {\\alpha} _ n \\\\\r\n& = \\sqrt{\\left( {\\alpha} _ n + {\\beta} _ n \\right)^2 -4 {\\alpha} _ n {\\beta} _ n} \\\\\r\n& = \\dfrac{\\sqrt{4 {b _ n}^2 -4 a _ n c _ n}}{a _ n} \\\\\r\n& = \\dfrac{2}{a _ n} \\sqrt{\\dfrac{D _ n}{4}} = \\dfrac{\\sqrt{D _ n}}{a _ n} \\quad ... [2]\r\n\\end{align}\\]\r\n(1)<\/strong> \u306e\u9014\u4e2d\u7d4c\u904e\u3088\u308a, \\(D _ n\\) \u306f \\(n\\) \u306b\u3088\u3089\u305a\u4e00\u5b9a\u3067\r\n\\[\r\nD _ n = D _ 1 = 6^2 -4 \\cdot 2 \\cdot 4 = 4\r\n\\]\r\n\u307e\u305f, \u6761\u4ef6\u3088\u308a \\(\\{ a _ n \\}\\) \u306f\u521d\u9805 \\(2\\) , \u516c\u6bd4 \\(4\\) \u306e\u7b49\u6bd4\u6570\u5217\u306a\u306e\u3067\r\n\\[\r\na _ n = 2 \\cdot 4^{n-1} = 2^{2n-1}\r\n\\]\r\n\u3053\u308c\u3089\u3092 [2] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\r\n\\text{P}{} _ n \\text{Q}{} _ n = \\left( \\dfrac{1}{4} \\right)^{n-1}\r\n\\]\r\n\u3088\u3063\u3066\u6c42\u3081\u308b\u548c\u306f\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {k=1}^n \\text{P}{} _ n \\text{Q}{} _ n & = \\dfrac{1 -\\left( \\frac{1}{4} \\right)^n}{1 -\\frac{1}{4}} \\\\\r\n& = \\underline{\\dfrac{4}{3} \\left\\{ 1 -\\left( \\dfrac{1}{4} \\right)^n \\right\\}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u6570\u5217 \\(\\{ a _ n \\}, \\{ b _ n \\}, \\{ c _ n \\}\\) \u3092 \\[\\begin{align} a _ 1 & = 2 , \\ a _ {n+1} = 4 a _ n \\ . \\\\ b _ … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n
\r\n\\(H _ k\\) \u304c \\(x\\) \u8ef8\u3068 \\(2\\) \u70b9\u3067\u4ea4\u308f\u308b, \u3059\u306a\u308f\u3061 \\(a _ k x^2 +2 b _ k x +c _ k = 0\\) \u306e\u5224\u5225\u5f0f \\(D _ k\\) \u306b\u3064\u3044\u3066\r\n\\[\r\n\\dfrac{D _ k}{4} = {b _ k}^2 -a _ k c _ k \\gt 0 \\quad ... [1]\r\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\(a _ {k+1} x^2 +2 b _ {k+1} x +c _ {k+1} = 0\\) \u306e\u5224\u5225\u5f0f \\(D _ {k+1}\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{D _ {k+1}}{4} & = {b _ {k+1}}^2 -a _ {k+1} c _ {k+1} \\\\\r\n& = \\left( b _ k +2 a _ k \\right)^2 -4 a _ k \\left( \\dfrac{c _ k}{4} +a _ k +b _ k \\right) \\\\\r\n& = {b _ k}^2 -a _ k c _ k \\\\\r\n& = \\dfrac{D _ k}{4} \\gt 0 \\quad ( \\ \\text{\u2235} \\ [1] \\ )\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \\(H _ {k+1}\\) \u3082 \\(x\\) \u8ef8\u3068 \\(2\\) \u70b9\u3067\u4ea4\u308f\u308b.<\/p><\/li>\r\n<\/ol>\r\n