(1)<\/strong>\u3000\\(\\dfrac{T _ {k+1}}{T _ k}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(\\displaystyle\\lim _ {n \\rightarrow \\infty} T _ k\\) \u3092 \\(S\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(\\text{A} _ {n}\\text{A} _ {n+1}\\) \u306e\u65b9\u7a0b\u5f0f\u306f\r\n\\[\\begin{align}\r\ny & = \\dfrac{{a _ {n}}^2-{a _ {n+1}}^2}{a _ {n}-a _ {n+1}} ( x-a _ {n} ) +{a _ {n}}^2 \\\\\r\n& = (a _ {n}+a _ {n+1}) x -a _ {n}a _ {n+1}\n\\end{align}\\]\r\n\\(C : \\ y=x^2\\) \u3088\u308a, \\(y'=2x\\) \u306a\u306e\u3067, \u70b9 \\(\\text{A} _ {n+2}\\) \u3067\u306e\u63a5\u7dda\u306e\u50be\u304d\u306f \\(2a _ {n+2}\\) . (2)<\/strong><\/p>\r\n (1)<\/strong> \u306e\u7d50\u679c\u3068 [2] \u3088\u308a\r\n\\[\r\nT _ k = \\left( \\dfrac{1}{8} \\right)^k T _ 1\n\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nS =\\dfrac{1}{6} \\left| a _ 1-a _ 2 \\right|^3 & = \\dfrac{4}{3} \\cdot \\dfrac{1}{8} \\left| a _ 1-a _ 2 \\right|^3 = \\dfrac{4}{3} T _ 1 \\\\\r\n\\text{\u2234} \\quad T _ 1 & = \\dfrac{3S}{4}\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} T _ k & = \\dfrac{3S}{4} \\cdot \\dfrac{1}{1 -\\frac{1}{8}} \\\\\r\n& = \\underline{\\dfrac{6S}{7}}\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u653e\u7269\u7dda \\(C : \\ y=x^2\\) \u4e0a\u306e\u70b9 \\(\\text{A} _ 1 \\left( a _ 1 , {a _ 1}^2 \\right)\\) , \\(\\text{A} _ 2 \\left( a _ 2 , {a _ … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
![\"\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/osaka_r_200901_01.png\" "\\"osaka_r_200901_01\\"")
\r\n
\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\na _ {n}a _ {n+1} & = 2a _ {n+2} \\\\\r\n\\text{\u2234} \\quad a _ {n+2} & = \\dfrac{a _ {n}+a _ {n+1}}{2} \\quad ... [1]\n\\end{align}\\]\r\n\\(\\text{A} _ {n}\\text{A} _ {n+1}\\) \u306e\u4e2d\u70b9 \\(\\text{M} _ n\\) \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\left( \\dfrac{a _ {n}+b}{2} , \\dfrac{{a _ {n}}^2+{a _ {n+1}}^2}{2} \\right)\n\\]\r\n\u306a\u306e\u3067, [1] \u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\nT _ k & = \\dfrac{1}{2} \\left\\{ \\dfrac{{a _ {n}}^2+{a _ {n+1}}^2}{2} +\\left( \\dfrac{a _ {n}+a _ {n+1}}{2} \\right)^2 \\right\\} \\left| a _ {n}-a _ {n+1} \\right| \\\\\r\n& = \\dfrac{1}{8} \\left| a _ {n}-a _ {n+1} \\right|^3 \\quad ... [2]\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\dfrac{T _ {k+1}}{T _ {k}} & = \\dfrac{\\left| a _ {n+1}-a _ {n+2} \\right|^3}{\\left| a _ {n}-a _ {n+1} \\right|^3} \\\\\r\n& = \\dfrac{\\left| a _ {n+1}-\\frac{a _ {n}+a _ {n+1}}{2} \\right|^3}{\\left| a _ {n}-a _ {n+1} \\right|^3} \\\\\r\n& = \\dfrac{\\left| \\frac{a _ {n+1}-a _ {n}}{2} \\right|^3}{\\left| a _ {n}-a _ {n+1} \\right|^3} = \\underline{\\dfrac{1}{8}}\n\\end{align}\\]\r\n