Given a real number sequence $\left\langle a _ { n } \right\rangle$ satisfying $a _ { 1 } = 1 , a _ { n + 1 } = \frac { 2 n + 1 } { 2 n - 1 } a _ { n } , n$ is a positive integer. Select the correct options.
(1) $a _ { 2 } = 3$
(2) $a _ { 4 } = 9$
(3) $\left\langle a _ { n } \right\rangle$ is a geometric sequence
(4) $\sum _ { n = 1 } ^ { 20 } a _ { n } = 400$
(5) $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n } = 2$