2.
已知二次函数\(f(x)= \dfrac {1}{3}x^{2}+ \dfrac {2}{3}x.\)数列\(\{a_{n}\}\)的前\(n\)项和为\(S_{n}\),点\((n,S_{n})(n∈N^{*})\)在二次函数\(y=f(x)\)的图象上.
\((\)Ⅰ\()\)求数列\(\{a_{n}\}\)的通项公式;
\((\)Ⅱ\()\)设\(b_{n}=a_{n}a_{n+1}\cos [(n+1)π](n∈N^{*})\),数列\(\{b_{n}\}\)的前\(n\)项和为\(T_{n}\),若\(T_{n}\geqslant tn^{2}\)对\(n∈N^{*}\)恒成立,求实数\(t\)的取值范围;
\((\)Ⅲ\()\)在数列\(\{a_{n}\}\)中是否存在这样一些项:\(a\;_{n_{1}}\),\(a\;_{n_{2}}\),\(a\;_{n_{3}}\),\(…\),\(a\;_{n_{k}}\)这些项都能够
构成以\(a_{1}\)为首项,\(q(0 < q < 5)\)为公比的等比数列\(\{a\;_{n_{k}}\}\)?若存在,写出\(n_{k}\)关于\(f(x)\)的表达式;若不存在,说明理由.