<多選題>已知實數數列 \( \langle a_n \rangle \) 滿足 \( a_1 = 1 \)，\( a_{n+1} = \frac{2n+1}{2n-1}a_n \)，\( n \)為正整數。試選出正確的選項。
(1) \( a_2 = 3 \)
(2) \( a_4 = 9 \)
(3) \( \langle a_n \rangle \) 為等比數列
(4) \( \sum_{n=1}^{20} a_n = 400 \)
(5) \( \lim\limits_{n \to \infty} \frac{a_n}{n} = 2 \)
答案
計算：\( a_2 = \frac{3}{1} \times 1 = 3 \)，\( a_3 = \frac{5}{3} \times 3 = 5 \)，\( a_4 = \frac{7}{5} \times 5 = 7 \)，\( a_5 = \frac{9}{7} \times 7 = 9 \)
(1) 正確，\( a_2=3 \)
(2) 錯誤，\( a_4=7 \)
(3) 錯誤，公比非常數
(4) \( a_n = 2n-1 \)，求和 \( \sum_{n=1}^{20} (2n-1) = 2\sum n - 20 = 2\times\frac{20\times21}{2}-20=420-20=400 \)，正確
(5) \( \lim_{n \to \infty} \frac{a_n}{n} = \lim_{n \to \infty} \frac{2n-1}{n} = 2 \)，正確
答案：(1)(4)(5) 報錯
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