已知\(\{a_{n}\}\)是各项均为正数的等比数列，且\(a_{1}+a_{2} =6\)，\(a_{1}a_{2}= a_{3}\)．

\((1)\)求数列\(\{a_{n}\}\)的通项公式；

\((2)\{b_{n}\}\)为各项非零的等差数列，其前\(n\)项和为\(S_{n}.\)已知\(S_{2n+1}=b_{n}b_{n+1}\)，求数列\(\left\{ \left. \dfrac{b_{n}}{a_{n}} \right. \right\}\)的前\(n\)项和\(T_{n}\)．

已知数列\(\{a_{n}\}\)是首项为\(1\)，公差为\(2\)的等差数列，数列\(\{b_{n}\}\)满足\(\dfrac{{{a}_{{1}}}}{{{b}_{{1}}}}+\dfrac{{{a}_{{2}}}}{{{b}_{{2}}}}+\dfrac{{{a}_{{3}}}}{{{b}_{{3}}}}+\ldots +\dfrac{{{a}_{n}}}{{{b}_{n}}}=\dfrac{{1}}{{{{2}}^{n}}}\)，若数列\(\{b_{n}\}\)的前\(n\)项和为\(S_{n}\)，则\(S_{5}=\)

\((1)\)已知各项均为正数的等比数列\(\{a_{n}\}\)满足\(a_{7}=a_{6}+2a_{5}\)，若存在两项\(a_{m}\)，\(a_{n}\)使得\( \sqrt{a_{m}a_{n}}=4a_{1}\)，则\( \dfrac{1}{m}+ \dfrac{4}{n}\)的最小值为_____________．

\((2)S_{n}\)为数列\(\{a_{n}\}\)的前\(n\)项和\(.\)已知\(a_{n} > 0\)，\(a\rlap{_{n}}{^{2}}+2a_{n}=4S_{n}+3\)．

\(②\)设\(b_{n}= \dfrac{1}{a_{n}a_{n+1}}\)，求数列\(\{b_{n}\}\)的前\(n\)项和．

已知\(\{a_{n}\}\)是一个公差大于\(0\)的等差数列，且满足\(a_{3}a_{5}=45\)，\(a_{2}+a_{6}=14\)．

\((1)\)求\(\{a_{n}\}\)的通项公式；

\((2)\)若数列\(\{b_{n}\}\)满足：\(\dfrac{{{b}_{1}}}{2}+\dfrac{{{b}_{2}}}{{{2}^{2}}}+\ldots +\dfrac{{{b}_{n}}}{{{2}^{n}}}={{a}_{n}}+1(n\in {{N}^{{*}}})\)，求\(\{b_{n}\}\)的前\(n\)项和．

已知数列\(\left\{ {{a}_{n}} \right\}\)中，\({{a}_{n}} > 0\)，\({{a}_{1}}=1\)，\({{a}_{n+2}}=\dfrac{1}{{{a}_{n}}+1}\)，\({{a}_{100}}={{a}_{96}}\)，则\({{a}_{2018}}+{{a}_{3}}=(\)   \()\)

已知\(f\left(n\right)= \dfrac{1}{n+1}+ \dfrac{1}{n+2}+ \dfrac{1}{n+3}+...+ \dfrac{1}{2n}\left(n∈{N}^{*}\right), \)那么\(f\left(n+1\right)-f\left(n\right) \)等于   \((\)   \()\)

已知数列\(\{b_{n}\}\)满足\(b_{1}=1\)，且\(16{b}_{n+1}={b}_{n}(n∈{N}^{*}) \)，设数列\(\left\{ \sqrt{{b}_{n}}\right\} \)的前\(n\)项和是\(T_{n}\) ．

\((1)\)比较\({{T}_{n+1}}^{2} \)与\({T}_{n}·{T}_{n+2} \)的大小；

\((2)\)若数列\(\{a_{n}\}\)的前\(n\)项和\(S_{n}=2n^{2}+2n+2\)，数列\(\{c_{n}\}\)满足\(c_{n}=a_{n}+\log _{d}b\)_{n\((d > 0,d\neq 1) \)}，求\(d\)的取值范围，使得数列\(\{c_{n}\}\)是递增数列．

等比数列中，分别是下表第一、二、三行中的某一个数，且中的任何两个数不在下表的同一列．

\((1)\)求数列\(\{a_{n}\}\)的通项公式；

\((2)\)若数列\(\{b_{n}\}\)满足：\(b_{n}=a_{n}+(-1)^{n}\ln a_{n}\)，求数列的\(2n\)前项和\(S2_{n}\)．