选择章节

已知等差数列\(\left\{ {{a}_{n}} \right\}\)的前\(n\)项和为\({{S}_{n}}\)，公差\(d > 0\)，且\({{a}_{2}}{{a}_{3}}=40\)，\({{a}_{1}}+{{a}_{4}}=13\)，公比为\(q(0 < q < 1)\)的等比数列\(\left\{ {{b}_{n}} \right\}\)中，\({{b}_{1}},{{b}_{3}},{{b}_{5}}\in \{\dfrac{1}{60},\dfrac{1}{32},\dfrac{1}{20},\dfrac{1}{8},\dfrac{1}{2}\}\)

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

\((\)Ⅱ\()\)若数列\(\left\{ {{c}_{n}} \right\}\)满足\({{c}_{2n-1}}={{a}_{n}},{{c}_{2n}}={{b}_{n}}\)，求数列\(\left\{ {{c}_{n}} \right\}\)的前\(2n\)项和\({{T}_{2n}}\)．

\(18.\)设数列\(\left\{ {{a}_{n}} \right\}\)的前\(n\)项和为\({{S}_{n}}\)，已知\({{a}_{1}}=2,{{a}_{2}}=8\)，\({{S}_{n+1}}+4{{S}_{n-1}}=5{{S}_{n}}\left( n\geqslant 2 \right)\)．

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

\((2)\)若\({{b}_{n}}={{\left( -1 \right)}^{n+1}}{lo}{{{g}}_{2}}{{a}_{n}}\)，求数列\(\left\{ {{b}_{n}} \right\}\)的前\(2n\)项和\({{T}_{2n}}\)。

设数列\({{a}_{n}}\)的前\(n\)项和为\({{S}_{n}}\)，且\({{S}_{n}}=2-\dfrac{1}{{{2}^{n-1}}},\{{{b}_{n}}\}\)为等差数列，且\({{a}_{1}}={{b}_{1}},{{a}_{2}}({{b}_{2}}-{{b}_{1}})={{a}_{1}}\)．

\((1)\)求数列\({{a}_{n}}\)和\(\{{{b}_{n}}\}\)的通项公式；

\((2)\)设\({{c}_{n}}=\dfrac{{{b}_{n}}}{{{a}_{n}}}\)，求数列\(\{{{c}_{n}}\}\)的前\(n\)项和\({{T}_{n}}\)．

已知正项等比数列\(\left\{ {{a}_{n}} \right\}\)的前\(n\)项和为\({{S}_{n}}\)，且\(2{{a}_{2}}={{S}_{2}}+\dfrac{1}{2}\)，\({{a}_{3}}=2\)．

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

\((\)Ⅱ\()\)若\({{b}_{n}}={lo}{{{g}}_{2}}{{a}_{n}}+3\)，数列\(\left\{ \dfrac{1}{{{b}_{n}}{{b}_{n+1}}} \right\}\)的前\(n\)项和为\({{T}_{n}}\)，求满足\({{T}_{n}} > \dfrac{1}{3}\)的正整数\(n\)的最小值．

已知数列\(\left\{ {{a}_{n}} \right\}\)是等差数列，数列\(\left\{ {{b}_{n}} \right\}\)是等比数列，且\({{b}_{2}}=3,{{b}_{3}}=9{{a}_{1}}={{b}_{1}},{{a}_{14}}={{b}_{4}}\)

\((1)\)求\(\left\{ {{a}_{n}} \right\}\)和\(\left\{ {{b}_{n}} \right\}\)的通项公式；

\((2)\)设\({{c}_{n}}={{a}_{n}}+{{b}_{n}}\)，求数列\(\left\{ {{c}_{n}} \right\}\)的前\(n\)项和\({{T}_{n}}\)．

已知数列\(\left\{ \sqrt{a_{n}}{-}n \right\}\)是等比数列，且\(a_{1}{=}9\)，\(a_{2}{=}36\)．

\((1)\)求数列\(\{ a_{n}\}\)的通项公式；

\((2)\)求数列\(\left\{ \sqrt{a_{n}} \right\}\)的前\(n\)项和\(S_{n}\)．

设数列\(\left\{ {{a}_{n}} \right\}\)的前\(n\)项和为\({{S}_{n}}\)\(.\)已知\(2{{S}_{n}}={{3}^{n}}+3\)．

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

\((\)Ⅱ\()\)若数列\(\left\{ {{b}_{n}} \right\}\)满足\({{a}_{n}}{{b}_{n}}={{\log }_{3}}{{a}_{n}}\)，求\(\left\{ {{b}_{n}} \right\}\)的前\(n\)项和\({{T}_{n}}\)．

已知数列\(\left\{ {{a}_{n}} \right\}\)满足\({{a}_{1}}=1\)，\(n{{a}_{n+1}}=3\left( n+1 \right){{a}_{n}}\)，设\({{b}_{n}}=\dfrac{{{a}_{n}}}{n}\)．

\((1)\)求\(b_{1}\)，\(b_{2}\)，\(b_{3}\)；

\((2)\)判断数列\(\left\{ {{b}_{n}} \right\}\)是否为等比数列，并说明理由；

\((3)\)求\(\left\{ {{a}_{n}} \right\}\)的通项公式．