1．
设正项数列\(\{{{a}_{n}}\}\)的前\(n\)项和\({{S}_{n}}\)，且满足\({{S}_{n}}=\dfrac{1}{2}a_{n}^{2}+\dfrac{n}{2}(n\in {{N}^{*}})\) ．
\((\)Ⅰ\()\)计算\({{a}_{1}},{{a}_{2}},{{a}_{3}}\)的值，猜想\(\{{{a}_{n}}\}\)的通项公式，并证明你的结论；
\((\)Ⅱ\()\)设\({{T}_{n}}\)是数列\(\{\dfrac{1}{a_{n}^{2}}\}\)的前\(n\)项和，证明：\({{T}_{n}} < \dfrac{4n}{2n+1}\) ．