知识点挑题 - 高中数学 - 优优班--学霸训练营

1．
已知数列\(\{a_{n}\}\)的前\(n\)项和为\(S_{n}\)，\(a_{1}= \dfrac {1}{2}\)，\(2a_{n+1}=S_{n}+1\)．
\((\)Ⅰ\()\)求\(a_{2}\)，\(a_{3}\)的值；
\((\)Ⅱ\()\)设\(b_{n}=2a_{n}-2n-1\)，求数列\(\{b_{n}\}\)的前\(n\)项和\(T_{n}\)．

难度：较难

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2．
已知数列\(\{a_{n}\}\)满足\(a_{7}=15\)，且点\((a_{n},a_{n+1})(n∈N^{*})\)在函数\(y=x+2\)的图象上．
\((1)\)求数列\(\{a_{n}\}\)的通项公式；
\((2)\)设\(b_{n}=3^{a_{n}}\)，求数列\(\{b_{n}\}\)的前\(n\)项和\(T_{n}\)．

难度：较易

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3．
已知数列\(\{a_{n}\}\)满足\(a_{n+2}-a_{n+1}=a_{n+1}-a_{n}\;(n∈N^{*})\)，且\(a_{3}+a_{7}=20\)，\(a_{2}+a_{5}=14\)．
\((1)\)求\(\{a_{n}\}\)的通项公式；
\((2)\)设\(b_{n}= \dfrac {1}{(a_{n}-1)\cdot (a_{n}+1)}\)，数列\(\{b_{n}\}\)的前\(n\)项和为\(T_{n}\)，求证：\(T_{n} < \dfrac {1}{2}\)．

难度：较易

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4．
已知数列\(\{a_{n}\}\)的前\(n\)项和\(S_{n}\)满足：\(S_{n}=2(a_{n}-1)\)，数列\(\{b_{n}\}\)满足：对任意\(n∈N*\)有\(a_{1}b_{1}+a_{2}b_{2}+…+a_{n}b_{n}=(n-1)⋅2^{n+1}+2\)．
\((1)\)求数列\(\{a_{n}\}\)与数列\(\{b_{n}\}\)的通项公式；
\((2)\)记\(c_{n}= \dfrac {b_{n}}{a_{n}}\)，数列\(\{c_{n}\}\)的前\(n\)项和为\(T_{n}\)，证明：当\(n\geqslant 6\)时，\(n|T_{n}-2| < 1\)．

难度：较易

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5．

定义\( \dfrac {n}{p_{1}+p_{2}+\cdots +p_{n}}\)为\(n\)个正数\(p_{1}\)，\(p_{2}…p_{n}\)的“平均倒数”\(.\)若已知数列\(\{a_{n}\}\)的前\(n\)项的“平均倒数”为\( \dfrac {1}{2n+1}\)，又\(b_{n}= \dfrac {a_{n}+1}{4}\)，则\( \dfrac {1}{b_{1}b_{2}}+ \dfrac {1}{b_{2}b_{3}}+…+ \dfrac {1}{b_{2017}b_{2018}}\)等于\((\)　　\()\)

A．\( \dfrac {2018}{2019}\)

B．\( \dfrac {2017}{2018}\)

C．\( \dfrac {2016}{2017}\)

D．\( \dfrac {2015}{2016}\)

难度：中等

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6．
等差数列\(\{a_{n}\}\)中，\(a_{2}=5\)，\(a_{1}+a_{4}=12\)，等比数列\(\{b_{n}\}\)的各项均为正数，且满足\(b_{n}b_{n+1}=2\;^{a_{n}}\)
\((\)Ⅰ\()\)求数列\(\{a_{n}\}\)的通项公式及数列\(\{b_{n}\}\)的公比\(q\)
\((\)Ⅱ\()\)求数列\(\{a_{n}+b_{n}\}\)的前\(n\)项和\(S_{n}\)．

难度：较易

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7．

设数列\(\{a_{n}\}\)的前\(n\)项和\(S_{n}\)，若\( \dfrac {a_{1}^{2}}{1^{2}}+ \dfrac {a_{2}^{2}}{2^{2}}+ \dfrac {a_{3}^{2}}{3^{2}}+…+ \dfrac {a_{n}^{2}}{n^{2}}=4n-4\)，且\(a_{n}\geqslant 0\)，则\(S_{100}\)等于\((\)　　\()\)

A．\(5048\)

B．\(5050\)

C．\(10098\)

D．\(10100\)

难度：较易

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8．
已知数列\(\{\{a_{n}\}\)满足\(a_{1}=1,a_{n+1}= \dfrac {a_{n}}{a_{n}+2}\)，\(b_{n+1}=(n-λ)( \dfrac {1}{a_{n}}+1)(n∈N^{*}),b_{1}=-λ\)．
\((1)\)求证：数列\(\{ \dfrac {1}{a_{n}}+1\}\)是等比数列；
\((2)\)若数列\(\{b_{n}\}\)是单调递增数列，求实数\(λ\)的取值范围．

难度：难

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9．
已知数列\(\{a_{n}\}\)的前\(n\)项和为\(S_{n}=3^{n}+1\)．
\((1)\)求数列\(\{a_{n}\}\)的通项公式；
\((2)\)令\(b_{n}= \dfrac {n}{a_{n}}\)，求数列\(\{b_{n}\}\)的前\(n\)项和\(T_{n}\)．

难度：较易

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10．
若数列\(\{a_{n}\}\)的前\(n\)项和\(S_{n}\)满足\(3S_{n}=1-a_{n}\)，\(n∈N^{*}\)．
\((1)\)求数列\(\{a_{n}\}\)的通项公式；
\((2)\)设\(b_{n}= \dfrac {1}{\log _{2}a_{n}\cdot \log _{2}a_{n+1}}(n∈N^{*})\)，求数列\(\{b_{n}\}\)的前\(n\)项和\(T_{n}\)．

难度：较易

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