17. (15 points) Given sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$ satisfying $a _ { 1 } = 2 , b _ { 1 } = 1 , a _ { n + 1 } = 2 a _ { n } \left( \mathrm { n } \in \mathrm { N } ^ { * } \right)$ , $b _ { 1 } + \frac { 1 } { 2 } b _ { 2 } + \frac { 1 } { 3 } b _ { 3 } + \cdots + \frac { 1 } { n } b _ { n } = b _ { n + 1 } - 1 \left( \mathrm { n } \in \mathrm { N } ^ { * } \right)$ . (1) Find $a _ { n }$ and $b _ { n }$ ; (2) Let $T _ { n }$ denote the sum of the first n terms of the sequence $\left\{ a _ { n } b _ { n } \right\}$ . Find $T _ { n }$ .
17. (15 points) Given sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$ satisfying $a _ { 1 } = 2 , b _ { 1 } = 1 , a _ { n + 1 } = 2 a _ { n } \left( \mathrm { n } \in \mathrm { N } ^ { * } \right)$ , $b _ { 1 } + \frac { 1 } { 2 } b _ { 2 } + \frac { 1 } { 3 } b _ { 3 } + \cdots + \frac { 1 } { n } b _ { n } = b _ { n + 1 } - 1 \left( \mathrm { n } \in \mathrm { N } ^ { * } \right)$ .\\
(1) Find $a _ { n }$ and $b _ { n }$ ;\\
(2) Let $T _ { n }$ denote the sum of the first n terms of the sequence $\left\{ a _ { n } b _ { n } \right\}$ . Find $T _ { n }$ .