已知等差数列\(\{a_{n}\}\)的前\(n\)项和为\(S_{n}\)，公差\(d≠0\)，\(a_{2}\)是\(a_{1}\)，\(a_{5}\)的等比中项，\(S_{5}=25.\)
\((1)\)求\(\{a_{n}\}\)的通项公式；
\((2)\)若数列\(\{b_{n}\}\)满足\({b}_{n}+{b}_{n+1}={2}^{\frac{{a}_{n}+1}{2}}\)，设\(\{b_{n}\}\)的前\(n\)项和为\(T_{n}\)，求\(T_{2n}.\)