曲线\(y=\sqrt{x}\)上点\({{P}_{1}},{{P}_{2}},\cdots {{P}_{n}}\)与\(x\)轴上点\({{Q}_{1}},{{Q}_{2}},\cdots {{Q}_{n}}\)构成一列正三角形,即\(\Delta O{{P}_{1}}{{Q}_{1}},\Delta {{Q}_{1}}{{P}_{2}}{{Q}_{2}},\cdots \Delta {{Q}_{n-1}}{{P}_{n}}{{Q}_{n}}\),设\(\Delta O{{P}_{1}}{{Q}_{1}}\)的边长为\({{a}_{1}}\),正三角形\(\Delta {{Q}_{n-1}}{{P}_{n}}{{Q}_{n}}\)的边长为\({{a}_{n}}\)
\((1)\)求\({{a}_{1}},{{a}_{2}}\)
\((2)\)求数列\(\left\{ {{a}_{n}} \right\}\)的通项公式
\((3)\)证明:\(\dfrac{1}{{{a}_{1}}^{2}}+\dfrac{1}{{{a}_{2}}^{2}}+\cdots +\dfrac{1}{{{a}_{n}}^{2}} < \dfrac{63}{16}\)