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:\({{T}_{n}} < \dfrac{4n}{2n+1}\) .