\((1)\)不等式\(\Delta ABD\)的解集为________．

\((2)\)若数列\(\left\{ {{a}_{n}} \right\}\)的前\(n\)项和为\({{S}_{n}}=\dfrac{2}{3}{{a}_{n}}+\dfrac{1}{3},\)则数列\(\left\{ {{a}_{n}} \right\}\)的通项公式是\({{a}_{n}}=\)_______．

\((3)\)在\(\Delta ABC\)中，角\(A,B,C\)的对边分别为\(a,b,c,\)且\({{a}^{2}}=b(b+c),\)则\(\dfrac{B}{A}{=}\)_______．

\((4)\)在平面四边形\(ABCD\)中，连接对角线\(BD\)，已知\(CD=9\)，\(BD=16\)，\(∠BDC=90^{\circ},\sin A= \dfrac{4}{5}, \)则对角线\(AC\)的最大值为________．

函数\(f\left( x \right)=\dfrac{{{\log }_{3}}\left( x+1 \right)}{x+1}\left( x > 0 \right)\)的图象上有一点列\({{P}_{n}}\left( {{x}_{n}},{{y}_{n}} \right)(n{∈}N_{{+}})\)，点\({{P}_{n}}\)在\(x\)轴上的射影是\({{Q}_{n}}\left( {{x}_{n}},0 \right)\)，且\({{x}_{n}}=3{{x}_{n-1}}+2\)\((\)\(n{\geqslant }2\)且\(n{∈}N\)\()\)，\({{x}_{1}}=2\)．

\((1)\)求出数列\(\left\{ {{x}_{n}} \right\}\)的通项公式；

\((2)\)对任意的正整数\(n\)，当\(m{∈}{[}{-}1{,}1{]}\)时，不等式\(3{{t}^{2}}-6mt+\dfrac{1}{3} > {{y}_{n}}\)恒成立，求实数\(t\)的取值范围；

\((3)\)设四边形\({{P}_{n}}{{Q}_{n}}{{Q}_{n+1}}{{P}_{n+1}}\)的面积是\({{S}_{n}}\)，求证：\(\dfrac{1}{S_{1}}{+}\dfrac{1}{{2S}_{2}}{+}{…}{+}\dfrac{1}{{nS}_{n}}{ < }\dfrac{5}{4}\)．

设\(S_{n}\)为正项数列\(\{ a_{n}\}\)的前\(n\)项和，\(a_{1}{=}2{,}S_{n{+}1}(S_{n{+}1}{-}2S_{n}{+}1){=}3S_{n}(S_{n}{+}1)\)，则\(a_{100}\)等于\(({  })\)

等比数列\(\{ a_{n}\}\)中，已知\(q{=}2{,}a_{2}{=}8\)，则\(a_{6}{=}\) ______ ．

已知各项都为正数的等比数列\(\left\{ {{a}_{n}} \right\}\)满足\(5{{a}_{1}}+4{{a}_{2}}={{a}_{3}}\)，且\({{a}_{1}}{{a}_{2}}={{a}_{3}}\)．

\((\)Ⅰ\()\)求数列\(\left\{ {{a}_{n}} \right\}\)的通项公式；

\((\)Ⅱ\()\)设\({{b}_{n}}={{\log }_{5}}{{a}_{n}}\)，且\({{S}_{n}}\)为数列\(\left\{ {{b}_{n}} \right\}\)的前\(n\)项和，求数列的\(\left\{ \dfrac{1}{{{S}_{n}}} \right\}\)的前\(n\)项和\({{T}_{n}}\)．