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职教组卷
  • 题型:解答题 题类:模拟题 难易度:中档

    已知公差不为\(0\)的等差数列\(\{a _{n} \}\)满足:\(a _{1} =1\),\(a _{2}\),\(a _{4}\),\(a _{8}\)成等比数列,数列\(\{b _{n} \}\)满足:\(b _{1} =1\),\(b _{n+1} = \dfrac {(b_{n}+n)b_{n}}{n}\),
    \((1)\)求数列\(\{a _{n} \}\)的通项公式;
    \((2)\)记数列\(c _{n} = \dfrac {1}{b_{n}+a_{n}}\),数列\(\{c _{n} \}\)的前\(n\)项和为\(T _{n}\),证明:\( \dfrac {1}{2}\leqslant T_{n} < 1\).
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