共50条信息
已知数列\(\{x_{n}\}\)满足\({{x}_{1}}=\dfrac{{1}}{{2}}\)，\({{x}_{n+1}}=\dfrac{{1}}{{1}+{{x}_{n}}}\)，\(n∈N^{*}\)．
\((1)\)猜想数列\(\{x_{2n}\}\)的单调性，并证明你的结论：
\((2)\)证明：\(|{{x}_{n+1}}-{{x}_{n}}|\leqslant \dfrac{1}{6}{{\left( \dfrac{2}{5} \right)}^{n-1}}\)．
\((2)\)已知双曲线\(C\)：\( \dfrac{{x}^{2}}{{a}^{2}}- \dfrac{{y}^{2}}{{b}^{2}}=1\left(a > 0,b > 0\right) \)的渐近线方程为\(y=\pm \dfrac{3}{4}x\)，且其右焦点为\((5,0)\)，则双曲线\(C\)的方程为______
\((3)\)椭圆\(\dfrac{{{x}^{2}}}{16}+\dfrac{{{y}^{2}}}{9}=1\)中，以点\(M(1,2)\)为中点的弦所在直线的方程为______
\((4)\)已知函数\(f(x)={{x}^{3}}-2{{x}^{2}}+ax+3\)在\(\left[ 1,2 \right]\)上单调递增，则实数\(a\)的取值范围为______
设\(f(x)=e^{x}(x+2x)\)，令\(f_{1}(x)=f{{"}}(x)\)，\(f_{n+1}(x)=f_{n}{{"}}(x)\)，若\(f_{n}(x)=e^{x}(A_{n}x^{2}+B_{n}x+C_{n})\)，则数列\(\left\{ \dfrac{1}{{{C}_{n}}} \right\}\)前\(n\)项和为\(S_{n}\)，当\(|{{S}_{n}}-1|\leqslant \dfrac{1}{2018}\)时，\(n\)的最小整数值为\((\) \()\)
设\(f_{0}(x){=}\cos x{,}f_{1}(x){=}f{{{{{"}}}}}_{0}(x){,}f_{2}(x){=}f{{{{{"}}}}}_{1}(x){,}f_{n{+}1}(x){=}f{{{{{"}}}}}_{n}(x)(n{∈}N)\)，则\(f_{2012}(x){=}({ })\)
已知整数对的序列如下：\(\left(1,1\right),\left(1,2\right),\left(2,1\right),\left(1,3\right),\left(2,2\right),\left(3,1\right),\left(1,4\right),\left(2,3\right),\left(3,2\right),\left(4,1\right),\left(1,5\right) \)则第\(57\)个数对是______．
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