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\(\{a_{n}\}\)的首项\(a_{1}=1\),前\(n\)项的和为\(S_{n}\),且满足\(2a_{n+1}+S_{n}=2(n∈N^{*})\),则满足\(\dfrac{1\mathrm{\ }001}{1\mathrm{\ }000} < \dfrac{S_{2n}}{S_{n}} < \dfrac{11}{10}\)的\(n\)的最大值为 \(.\)
数列\(\{a_{n}\}\)满足\({{a}_{n+1}}=\dfrac{{{a}_{n}}}{2{{a}_{n}}+1}\),\({{a}_{3}}=\dfrac{1}{5}\)则\(a_{1}=\)________.
数列\(\{a_{n}\}\)满足\(a_{1}=1\),\(a_{n+1}=(n^{2}+n-λ)a_{n}(n=1,2,…)\),\(λ\)是常数.
\((1)\)当\(a_{2}=-1\)时,求\(λ\)及\(a_{3}\)的值;
\((2)\)是否存在实数\(λ\)使数列\(\{a_{n}\}\)为等差数列?若存在,求出\(λ\)及数列\(\{a_{n}\}\)的通项公式;若不存在,请说明理由.
已知数列\(\{a_{n}\}\)满足\(a_{n+2}=a_{n+1}-a_{n}\),且\(a_{1}=2\),\(a_{2}=3\),则\(a_{2018}\)的值为________.
在数列\(\{{{a}_{n}}\}\)中,\({{a}_{1}}=4,{{a}_{n+1}}-1=3({{a}_{n}}-1)\) ,则数列\(\left\{ {{a}_{n}} \right\}\)的通项公式\({{a}_{n}}=\) ______.
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