$301. $ 已知\(\displaystyle a_n = \frac{3^n+1}{4}\)且\(S_n = \sum \limits_{k=1}^n a_k\)，证明：当\(n \ge 2\)时，\(\displaystyle \frac{8}{3}(n+1)S_n > (n+1)C_{n+1}^02^n + n C_{n+1}^1 2^{n-1} + \cdots + C_{n+1}^n 2^0\)。
$301. $ 已知\(\displaystyle a_n = \frac{3^n+1}{4}\)且\(S_n = \sum \limits_{k=1}^n a_k\)，证明：当\(n \ge 2\)时，\(\displaystyle \frac{8}{3}(n+1)S_n > (n+1)C_{n+1}^02^n + n C_{n+1}^1 2^{n-1} + \cdots + C_{n+1}^n 2^0\)。