A lottery game has a single-play winning probability of 0.1, and each play is an independent event. For each positive integer $n$, let $p_{n}$ be the probability of winning at least once in $n$ plays of this game. Select the correct options. (1) $p_{n+1} > p_{n}$ (2) $p_{3} = 0.3$ (3) $\langle p_{n} \rangle$ is an arithmetic sequence (4) Playing this game two or more times, the probability of not winning on the first play and winning on the second play equals $p_{2} - p_{1}$ (5) When playing this game $n$ times with $n \geq 2$, the probability of winning at least 2 times equals $2p_{n}$
A lottery game has a single-play winning probability of 0.1, and each play is an independent event. For each positive integer $n$, let $p_{n}$ be the probability of winning at least once in $n$ plays of this game. Select the correct options.\\
(1) $p_{n+1} > p_{n}$\\
(2) $p_{3} = 0.3$\\
(3) $\langle p_{n} \rangle$ is an arithmetic sequence\\
(4) Playing this game two or more times, the probability of not winning on the first play and winning on the second play equals $p_{2} - p_{1}$\\
(5) When playing this game $n$ times with $n \geq 2$, the probability of winning at least 2 times equals $2p_{n}$