共50条信息
在数列\(\left\{ {{a}_{n}} \right\}\)中,\({{a}_{{1}}}{+2}{{a}_{2}}+{{2}^{2}}{{a}_{3}}+\cdots +{{2}^{n-1}}{{a}_{n}}=(n\cdot {{2}^{n}}-{{2}^{n}}+1)\ t\)对任意\(n\in {{N}^{*}}\)成立,其中常数\(t > 0.\)若关于\(n\)的不等式\(\dfrac{1}{{{a}_{2}}}+\dfrac{1}{{{a}_{4}}}+\dfrac{1}{{{a}_{8}}}+\cdots +\dfrac{1}{{{a}_{{{2}^{n}}}}} > \dfrac{m}{{{a}_{1}}}\)的解集为\(\{n|n\geqslant 4,n\in {{N}^{*}}\}\),则实数\(m\)的取值范围是 .
已知数列\(\left\{ {{a}_{n}} \right\}\)的通项公式\({{a}_{n}}=-n+t\),数列\(\left\{ {{b}_{n}} \right\}\)的通项公式\({{b}_{n}}={{2}^{n}}\),设数列\(\left\{ {{c}_{n}} \right\}\)满足\({{c}_{n}}=\dfrac{{{a}_{n}}+{{b}_{n}}}{2}+\dfrac{\left| {{a}_{n}}-{{b}_{n}} \right|}{2}\),且\({{c}_{n}}\geqslant {{c}_{3}}\left( n\in {{N}^{{*}}} \right)\),则实数\(t\)的取值范围是________________
已知数列\(\left\{ {{a}_{n}} \right\}\)满足:\({{a}_{1}}=1\),\({{a}_{n+1}}=\dfrac{{{a}_{n}}}{{{a}_{n}}+2}\) \(\left( n\in {{N}^{*}} \right).\)若\({{b}_{n+1}}=\left( n-2\lambda \right)\cdot \left( \dfrac{1}{{{a}_{n}}}+1 \right)\) \(\left( n\in {{N}^{*}} \right)\),\({{b}_{1}}=-\lambda \),且数列\(\left\{ {{b}_{n}} \right\}\)是单调递增数列,则实数\(\lambda \)的取值范围是____。
已知数列\(\{a_{n}\}\)满足\({{a}_{n+1}}=\dfrac{1}{1-{{a}_{n}}}(n∈N*)\),\(a_{8}=2\),则\(a_{1}\)的值为\((\) \()\)
已知首项为\(\dfrac{3}{2}\)的等比数列\(\{{{a}_{n}}\}\)的前\(n\)项和为\({{S}_{n}}\),\((n\in {{N}^{*}})\),且\(-2{{S}_{2}},{{S}_{3}},4{{S}_{4}}\)成等差数列,
\((\)Ⅰ\()\)求数列\(\{{{a}_{n}}\}\)的通项公式;
\((\)Ⅱ\()\)求\({{S}_{n}}(n\in {{N}^{*}})\)的最值.
数列\(\left\{ {{a}_{n}} \right\}\)满足\({a}_{n+1}=\begin{cases}2{a}_{n},(0\leqslant {a}_{n} < \dfrac{1}{2}) \\ 2{a}_{n}-1,( \dfrac{1}{2}\leqslant {a}_{n} < 1)\end{cases} \),若\({{a}_{1}}=\dfrac{3}{5}\),则\({{a}_{2014}}=\)( )
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