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#a2e5dde9-8a1c-4630-8e52-9ca96e30d54e中等解答题数列求通项方法数列

27.已知数列{an}\{a_{n}\}的前nn项和为SnS_{n}a1=1a_{1}=1Sn+1=2Sn+2n+1{S}_{n+1}=2{S}_{n}+{2}^{n+1}nNn\in N^{*}. (1)求数列{an}\{a_{n}\}的通项公式; (2)设bn=Sn3n{b}_{n}=\frac{{S}_{n}}{{3}^{n}}{bn}\{b_{n}\}的前nn项和为TnT_{n},若对任意的正整数nn,不等式Tn>m2m+727{T}_{n}>\frac{{m}^{2}-m+7}{27}恒成立,求实数mm的取值范围.