题型：解答题 题类：期末考试 难易度：较易
新年份：2021
已知等比数列\(\left\{\begin{array}{l}a_{n}\end{array}\right\}\)的公比\(q=3\)，并且满足\(a_{2}\)，\(a_{3}+18\)，\(a_{4}\)成等差数列.
\((1)\)求数列\(\left\{\begin{array}{l}a_{n}\end{array}\right\}\)的通项公式；
\((2)\)设数列\(\left\{\begin{array}{l}b_{n}\end{array}\right\}\)满足\(b_{n}=\dfrac{1}{a_{n}}+\log_{3}a_{n}\)，记\(S_{n}\)为数列\(\left\{\begin{array}{l}b_{n}\end{array}\right\}\)的前\(n\)项和，求使\(2S_{n}-n^{2}>20\)成立的正整数\(n\)的最小值.
