已知正项数列\(\{a_{n}\}\)满足\(a_{1}=3\)，\(\dfrac{a_{n}+a_{n-1}}{a_{n}-a_{n-1}}=2n(n\in N^{*},n\geqslant 2).\)
\((Ⅰ)\)写出\(a_{2}\)，\(a_{3}\)，并证明数列\(\{a_{n}\}\)是等差数列；
\((Ⅱ)\)设数列\(\{b_{n}\}\)满足\(b_{1}=2\)，\(b_{n+1}b_{n}-b_{n}^{2}=a_{n}(n\in N^{*})\)，求证：\(b_{1}+b_{2}+b_{3}+⋯+b_{n}\geqslant\dfrac{n(n+3)}{2}.\)