已知在数列\(\{a_{n}\}\)中，\(a_{1}=1\)，\(a_{n+1}=2a_{n}+n-1(n∈N*)\)，则其前\(n\)项和\(S_{n}=\)________．

等差数列\(\{a_{n}\}\)中，\(2a_{1}+3a_{2}=11\)，\(2a_{3}+a_{6}-4\)，其前\(n\)项和为\(S_{n}\)．

\((1)\)求数列\(\{a_{n}\}\)的通项公式；

\((2)\)设数列\(\{b_{n}\}\)满足\({{b}_{n}}=\dfrac{1}{{{S}_{n+1}}-1}\)，其前\(n\)项和为\(T_{n}\)，求证：\({{T}_{n}} < \dfrac{3}{4}(n∈N^{*})\)

\(( 1 )\)已知向量\(\overrightarrow{a},\overrightarrow{b}\)，满足\(\overrightarrow{a}=\left( 1,3 \right)\)，\(\left( \overrightarrow{a}+\overrightarrow{b} \right)\bot \left( \overrightarrow{a}-\overrightarrow{b} \right)\)，则\(\left| \overrightarrow{b} \right|=\)______．

\(( 2 )\)已知实数\(x,y\)满足\(\begin{cases} & x\leqslant 3 \\ & x+y-3\geqslant 0 \\ & x-y+1\geqslant 0 \\ \end{cases}\)，则\({{x}^{2}}+{{y}^{2}}\)的最小值是     ．

\(( 3 )\)已知圆\(O:{{x}^{2}}+{{y}^{2}}=1.\)圆\({O}{{'}}\)与圆\(O\)关于直线\(x+y-2=0\)对称，则圆\({O}{{'}}\)的方程是__________．

\(( 4 )\)已知数列\(\left\{ a{}_{n} \right\},\left\{ {{b}_{n}} \right\}\)满足\(b{}_{n}=\log {}_{2}a{}_{n},n\in {{N}^{*}}\)，其中\(\left\{ {{b}_{n}} \right\}\)是等差数列，且\({{a}_{9}}{{a}_{2009}}=\dfrac{1}{4}.\)则\({{b}_{1}}+{{b}_{2}}+{{b}_{3}}+\cdot \cdot \cdot +{{b}_{2017}}=\)__________．

设数列\(\left\{ {{a}_{n}} \right\}\)的前\(n\)项的和为\({{S}_{n}}\)，点\((n,{{S}_{n}})\)在函数\(f(x)=2{{x}^{2}}\)的图象上，数列\(\left\{ {{b}_{n}} \right\}\)满足：\({{b}_{1}}={{a}_{1}},{{b}_{n+1}}({{a}_{n+1}}-{{a}_{n}})={{b}_{n}}.\)其中\(n\in {{N}^{*}}\)．

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

\((\)Ⅱ\()\)设\({{c}_{n}}=\dfrac{{{a}_{n}}}{{{b}_{n}}}\)，求证：数列\(\left\{ {{c}_{n}} \right\}\)的前\(n\)项的和\({{T}_{n}} > \dfrac{5}{9}(n\in {{N}^{*}}).\)

设数列\(\left\{ {{a}_{n}} \right\}\)的前\(n\)项和为\({{S}_{n}}\)，已知\(\dfrac{{{S}_{n}}}{2}={{a}_{n}}-{{2}^{n}} (\)\(n\)\(∈N*)\)．

\((1)\)求\({{a}_{1}}\)的值，若\({{a}_{n}}={{2}^{n}}{{c}_{n}}\)，证明数列\(\{{{c}_{n}}\}\)是等差数列；

\((2)\)设\({{b}_{n}}={{\log }_{2}}{{a}_{n}}-{{\log }_{2}}(n+1)\)，数列\(\{\dfrac{1}{{{b}_{n}}}\}\)的前\(n\)项和为\({{B}_{n}}\)，若存在整数\(m\)，使对任意\(n\)\(∈\)\(N\)\(*\)且\(n\)\(\geqslant 2\)，都有\({{B}_{3n}}-{{B}_{n}} > \dfrac{m}{20}\)成立，求\(m\)的最大值．

已知公差不为零的等差数列\(\left\{{a}_{n}\right\} \)中，\({{a}_{3}}=7\)，且\({{a}_{1}},{{a}_{4}},{{a}_{13}}\)成等比数列

\((1)\)求数列\(\left\{{a}_{n}\right\} \)的通项公式      

\((2)\)令\({{b}_{n}}=\dfrac{1}{{{a}^{2}}_{n}-1}\) \((n∈N^{*})\)，求数列\(\left\{{b}_{n}\right\} \)的前\(n\)项和\({{S}_{n}}\)

已知数列\(\{b_{n}\}\)是首项为\(2\)且公比为\(q\)的等比数列，数列\(\{a_{n}\}\)满足\(a_{1}=3q\)，\(a_{n+1}-qb_{n+1}=a_{n}-qb_{n}(n∈N^{*}).\)

\((1)\)求数列\(\{a_{n}\}\)的通项公式；

\((2)\)若\(q= \dfrac{1}{2} \)，数列\(\{b_{n}\}\)前\(n\)项和为\(S_{n}\)，求所有满足等式\( \dfrac{{S}_{n}-m}{{S}_{n+1}-m}= \dfrac{1}{{b}_{m}+1} \)成立的正整数\(m\)，\(n\)；

\((3)\)若\(q < 0\)，且对任意\(m\)，\(n∈N^{*}\)，都有\( \dfrac{{a}_{m}}{{a}_{n}}∈\left( \dfrac{1}{6},6\right) \)，求实数\(q\)的取值范围．