设数列\(\{a_{n}\}\)为等差数列，数列\(\{b_{n}\}\)为等比数列\(.\)若\(a_{1} < a_{2}\)，\(b_{1} < b_{2}\)，且\({b}_{i}={{a}_{i}}^{2}(i=1,2,3) \)，则数列\(\{b_{n}\}\)的公比为________．

设\({{S}_{n}}\)是等差数列\(\left\{ a{}_{n} \right\}\)的前\(n\)项和，若\( \dfrac{{a}_{5}}{{a}_{3}}= \dfrac{5}{9}, \)则\( \dfrac{{S}_{9}}{{S}_{5}} =(\)   \()\)

已知方程\(\left({x}^{2}-2x+m\right)\left({x}^{2}-2x+n\right)=0 \)的四个根组成一个首项为\( \dfrac{1}{4} \)的等差数列，则\(\left|m-n\right| \)等于

\((1)\)不等式\(\dfrac{1}{x} < 1\)的解集是________．

\((2)\)已知\(a\)，\(b\)是互异的正数，\(A\)是\(a\)，\(b\)的等差中项，\(G\)是\(a\)，\(b\)的正的等比中项，则\(A\)________\(G( > , < ,\geqslant ,\leqslant \)选填其中一个\()\)．

\((3)\)已知\(\sin (60{}^\circ +\alpha )=\dfrac{5}{13}\)，\(30^{\circ} < a < 120^{\circ}\)，则\(\cos α=\)________．

\((4)\)如图在正方体\(ABCD—A_{1}B_{1}C_{1}D_{1}\)中，给出以下结论

\(①A_{1}C_{1}\)与平面\(A_{1}B_{1}CD\)成\(45^{\circ}\)角；

\(②CD_{1}\)与\(BC_{1}\)成\(60^{\circ}\)角；

\(③{{V}_{B1}}_{-{{A}_{1}}B{{C}_{1}}}=\dfrac{1}{2}{{V}_{B}}{{_{1}}_{-A{{D}_{1}}C}}\)；

\(④\)正方体的内切球，与各条棱相切的球，外接球的表而积之比为\(1︰2︰3\)其中正确的结论序号是________\(.(\)写出所有正确结论的序号\()\)

已知数列\(\left\{ {{a}_{n}} \right\}\)，\({{a}_{n}} > 0\)，其前\(n\)项和\({{S}_{n}}\)满足\({{S}_{n}}=2{{a}_{n}}-{{2}^{n+1}}\)，其中\(n\in N*\)．

\((\)Ⅰ\()\)设\({{b}_{n}}=\dfrac{{{a}_{n}}}{{{2}^{n}}}\)，证明：数列\(\left\{ {{b}_{n}} \right\}\)是等差数列；

\((\)Ⅱ\()\)设\({{c}_{n}}={{b}_{n}}\cdot {{2}^{-n}}\)，\({{T}_{n}}\)为数列\(\left\{ {{c}_{n}} \right\}\)的前\(n\)项和，求证：\({{T}_{n}} < 3\)；

\((\)Ⅲ\()\)设\({{d}_{n}}={{4}^{n}}+{{(-1)}^{n-1}}\lambda \cdot {{2}^{{{b}_{n}}}}(\lambda \)为非零整数，\(n\in N*)\)，试确定\(\lambda \)的值，使得对任意\(n\in N*\)，都有\({{d}_{n+1}} > {{d}_{n}}\)成立．