\((1)\)已知\(\sin α= \dfrac{3}{5}\)，\(α∈( \dfrac{π}{2},π)\)，则\(\cos \alpha =\)________，\( \dfrac{\cos 2α}{ \sqrt{2}\sin (α+ \dfrac{π}{4})}=\)________．

\((2)\)已知数列\(\{a_{n}\}\)的首项为\(1\)，数列\(\{b_{n}\}\)为等比数列且\(b_{n}= \dfrac{a_{n+1}}{a_{n}}\)，若\(b_{10}·b=2\)，则\({{b}_{7}}{{b}_{14}}=\)_____，\(a_{21}=\)________．

\((3)\)计算：\(\tan 20^{\circ}+\tan 40^{\circ}+ \sqrt{3}\tan 20^{\circ}\tan 40^{\circ}=\)________，\( \dfrac{ \sqrt{3}\tan 12^{\circ}-3}{(4\cos ^{2}12^{\circ}-2)\sin 12^{\circ}}=\)________．

\((4)\)数列\(\left\{ {{a}_{n}} \right\}\)满足\({{a}_{1}}=1\)，\(\sqrt{\dfrac{1}{{{a}_{n}}^{2}}+2}=\dfrac{1}{{{a}_{n+1}}}\left( n\in {{N}^{*}} \right)\)，记\({{b}_{n}}={{a}_{n}}^{2}\)，则数列\(\left\{ {{a}_{n}} \right\}\)的通项公式\({{a}_{n}}=\)____________，数列\(\left\{ {{b}_{n}}{{b}_{n+1}} \right\}\)前\(n\)项和\({{S}_{n}}=\)___________．

\((5)\)在\(200 m\)高的山顶上，测得山下一塔顶与塔底的俯角分别为\(30^{\circ}\)与\(60^{\circ}\)，则塔高是_____\(m\)．

\((6)\)若\(\sin \alpha +\sin \beta =\dfrac{\sqrt{2}}{2},\)则\(\cos \alpha +\cos \beta \)的取值范围_____．

\((7)\)设数列\({{a}_{n}}\)满足：\({{a}_{1}}=\sqrt{3}\)，\({{a}_{n+1}}=\left[ {{a}_{n}} \right]+\dfrac{1}{\left\{ {{a}_{n}} \right\}}\)，其中，\(\left[ {{a}_{n}} \right]\)、\(\left\{ a{}_{n} \right\}\)分别表示正数\({{a}_{n}}\)的整数部分、小数部分，则\({{a}_{2018}}=\)_____．

已知正项数列\(\{a_{n}\}\)的前\(n\)项和为\(S_{n}\)，对任意\(n∈N^{*}\)，点\(\left( \left. a_{n},S_{n} \right. \right)\)都在函数\(f\left( \left. x \right. \right)= \dfrac{1}{2}x^{2}+ \dfrac{1}{2}x\)的图象上．

\((1)\)求数列\(\{a_{n}\}\)的首项\(a_{1}\)和通项公式\(a_{n}\)；

\((2)\)若数列\(\left\{ \left. b_{n} \right. \right\}\)满足\(\log _{2}b_{n}=n+\log _{2}\left( \left. 2a_{n}-1 \right. \right)\left( \left. n∈N^{*} \right. \right)\)，求数列\(\left\{ \left. b_{n} \right. \right\}\)的前\(n\)项和\(T_{n}\)；

\((3)\)已知数列\(\left\{ \left. c_{n} \right. \right\}\)满足\(c_{n}= \dfrac{4n-6}{T_{n}-6}- \dfrac{1}{a_{n}a_{n+1}}\left( \left. n∈N^{*} \right. \right).\)若对任意\(n∈N^{*}\)，存在\(x_{0}∈\left[ \left. - \dfrac{1}{2}, \dfrac{1}{2} \right. \right]\)，使得\(c_{1}+c_{2}+…+c_{n}\leqslant f(x)-a\)成立，求实数\(a\)的取值范围．

已知\(\triangle ABC\)的三个内角\(A\)，\(B\)，\(C\)成等差数列，角\(B\)所对的边\(b= \sqrt{3}\)，且函数\(f(x)=2 \sqrt{3}\sin ^{2}x+2\sin x\cos x- \sqrt{3}\)在\(x=A\)处取得最大值，则\(\triangle ABC\)的面积是________  

公差不为零的等差数列\(\{a_{n}\}\)中，\(a_{1}\)，\(a_{2}\)，\(a_{5}\)成等比数列，且该数列的前\(10\)项和为\(100.\)数列\(\{b_{n}\}\)的前\(n\)项和为\({S}_{n} \)，且满足\(S_{n}=2b_{n}-1,\;\;n∈N^{*}\)．
\((I)\)求数列\(\{a_{n}\}\)，\(\{b_{n}\}\)的通项公式；

\((\)Ⅱ\(){C}_{n}={a}_{n}+{\log }_{ \sqrt{2}}{b}_{n}, \)数列\(\left\{{C}_{n}\right\} \)的前\(n \)项和为\({T}_{n} \)，若数列\(\left\{{d}_{n}\right\} \)为等差数列，且\({d}_{n}= \dfrac{{T}_{n}}{n-c}\left(其中,c\neq 0\right) \)．

\((i)\)求非零常数\(c ;(ii)\)若\(f(n)=\; \dfrac{{d}_{n}}{(n+36){d}_{(n+1)}}(n∈{N}^{*}), \)求数列\(\left\{f\left(n\right)\right\} \)的最大项的值．

已知无穷数列\(\left\{{a}_{n}\right\} \)的各项都是正数，其前\(n\)项和为\({S}_{n} \)，且满足：\({a}_{1}=a \)，\(r{S}_{n}={a}_{n}{a}_{n+1}-1 \)，其中\(a\neq 1 \)，常数\(r∈N \)．

\((1)\)求证：\({a}_{n+2}-{a}_{n} \)是一个定值；

\((2)\)若数列\(\left\{{a}_{n}\right\} \)是一个周期数列\((\)存在正整数\(T\)，使得对任意\(n∈{N}^{*} \)，都有\({a}_{n+T}={a}_{n} \)成立，则称\(\left\{{a}_{n}\right\} \)为周期数列，\(T\)为它的一个周期\()\)，求该数列的最小周期；

\((3)\)若数列\(\left\{{a}_{n}\right\} \)是各项均为有理数的等差数列，\({c}_{n}=2·{3}^{n-1} (n∈{N}^{*} )\)，问：数列\({c}_{n} \)中的所有项是否都是数列\(\left\{{a}_{n}\right\} \)中的项？若是，请说明理由；若不是，请举出反例．

\((1)\)已知\(| \overset{→}{a}|=1,| \overset{→}{b}|=2 \)，\(| \overset{→}{a}-2 \overset{→}{b}|= \sqrt{13} \)，则\( \overset{→}{a} \)与\( \overset{→}{b} \)的夹角为______．

\((2)\)已知数列\(\left\{ {{a}_{n}} \right\}\)中，\({{a}_{3}}=2,{{a}_{7}}=1\)，且数列\(\left\{ \dfrac{1}{{{a}_{n}}+1} \right\}\)是等差数列，则\({{a}_{11}}=\)       

\((3)\)已知函数\(f\)\((\)\(x\)\()=\begin{cases}2\sin πx,x < 1 \\ f(x- \dfrac{2}{3}),x\geqslant 1\end{cases} \)，则\( \dfrac{f(2)}{f(- \dfrac{1}{6})}= = \)______．

\((4)\)在数列\(\{\)\(a_{n}\)\(\}\)中，若\(a_{n}\)\({\,\!}^{2}-\)\(a_{n}\)\({\,\!}_{-1}^{2}=\)\(p\)\((\)\(n\)\(\geqslant 2\)，\(n\)\(∈N^{×}\)，\(p\)为常数\()\)，则称\(\{\)\(a_{n}\)\(\}\)为“等方差数列”，下列是对“等方差数列”的判断；

\(①\)若\(\{\)\(a_{n}\)\(\}\)是等方差数列，则\(\{\)\(a_{n}\)\({\,\!}^{2}\}\)是等差数列；

\(②\{(-1)\)\({\,\!}^{n}\)\(\}\)是等方差数列；

\(③\)若\(\{\)\(a_{n}\)\(\}\)是等方差数列，则\(\{\)\(a\) \(\}(\)\(k\)\(∈N_{+}\)，\(k\)为常数\()\)也是等方差数列；

\(④\)若\(\{\)\(a_{n}\)\(\}\)既是等方差数列，又是等差数列，则该数列为常数列．

其中正确命题序号为______\(.(\)将所有正确的命题序号填在横线上\()\)

已知数列\(\left\{ {{a}_{n}} \right\}\)，\(\left\{ {{b}_{n}} \right\}\)满足\({{a}_{1}}=\dfrac{1}{4},{{a}_{n}}+{{b}_{n}}=1,{{b}_{n+1}}=\dfrac{{{b}_{n}}}{(1-{{a}_{n}})(1+{{a}_{n}})}\)．

 \((\)Ⅰ\()\)求\({{b}_{1}},{{b}_{2}},{{b}_{3}},{{b}_{4}}\)；

 \((\)Ⅱ\()\)设\({{c}_{n}}=\dfrac{1}{{{b}_{n}}-1}\)，证明数列\(\left\{ {{c}_{n}} \right\}\)是等差数列；

 \((\)Ⅲ\()\)设\({{S}_{n}}={{a}_{1}}{{a}_{2}}+{{a}_{2}}{{a}_{3}}+{{a}_{3}}{{a}_{4}}+...+{{a}_{n}}{{a}_{n+1}}\)，不等式\(4a{{S}_{n}} < {{b}_{n}}\)恒成立时，求实 数\(a\)的取值范围．

把满足条件\(T\)的数列\(\{\)\(a_{n}\)\(\}\)构成的集合记为\(M\)，其中条件\(T\)：任意\(m\)，\(n\)\(∈N*\)，都有\(a_{n}\)\({\,\!}_{+}\)\({\,\!}_{m}\)\(\geqslant \)\(a_{n}\)\(+\)\(a\)\({\,\!}_{m}\)．

 \((1)\)已知等差数列\(\{\)\(a_{n}\)\(\}\)的前\(n\)项的和为\(S_{n}\)，且\(a\)\({\,\!}_{2}=3\)，\(a\)\({\,\!}_{3}+\)\(a\)\({\,\!}_{5}=10\)，求证：\(\{\)\(S_{n}\)\(\}\)\(M\)；

 \((2)\)已知数列\(\{\)\(a_{n}\)\(\}\)是各项均为正数的等比数列，且\(\{\)\(a_{n}\)\(\}\)\(M\)，求数列\(\{\)\(a_{n}\)\(\}\)公比\(q\)的最小值；

 \((3)\)已知数列\(\{\)\(a_{n}\)\(\}\)的各项均为正整数，且\(\{\)\(a_{n}\)\(\}\)\(M\)\(.\)设\(b_{n}\)\(=\)\(a_{a}\)\({\,\!}^{n}\)，\(c_{n}\)\(=\)\(a\)\({\,\!}_{1+}\)\({\,\!}_{a}\)\({\,\!}^{n}\)，若数列\(\{\)\(b\)\({\,\!}_{n}\)\(\}\)，\(\{\)\(c_{n}\)\(\}\)均为等差数列，求证：数列\(\{\)\(b\)\({\,\!}_{n}\)\(\}\)，\(\{\)\(c_{n}\)\(\}\)有相同的公差．

已知数列\(\{{{a}_{n}}\}\)满足\({{a}_{n}}=2{{a}_{n-1}}+{{2}^{n}}-1(n\in {{\mathbf{N}}_{+}}\)，且\(n\geqslant 2)\)，\({{a}_{4}}={81}\)．

\((\)Ⅰ\()\)求数列的前三项\({{a}_{1}}\)，\({{a}_{2}}\)，\({{a}_{3}}\)；

\((\)Ⅱ\()\)数列\(\left\{ \dfrac{{{a}_{n}}+p}{{{2}^{n}}} \right\}\)为等差数列，求实数\(p\)的值；

\((\)Ⅲ\()\)求数列\(\{{{a}_{n}}\}\)的前\(n\)项和\({{S}_{n}}\)．