gaokao 2023 Q17

gaokao · China · national-A-science 12 marks Sequences and series, recurrence and convergence Closed-form expression derivation
In the sequence $\left\{ a_{n} \right\}$ , $a_{2} = 1$ . Let $S_{n}$ be the sum of the first $n$ terms of $\left\{ a_{n} \right\}$ . $2S_{n} = na_{n}$ .
(1) Find the general term formula for $\left\{ a_{n} \right\}$ ;
(2) Find the sum $T_{n}$ of the first $n$ terms of the sequence $\left\{ \frac{a_{n} + 1}{2^{n}} \right\}$ .
In the sequence $\left\{ a_{n} \right\}$ , $a_{2} = 1$ . Let $S_{n}$ be the sum of the first $n$ terms of $\left\{ a_{n} \right\}$ . $2S_{n} = na_{n}$ .

(1) Find the general term formula for $\left\{ a_{n} \right\}$ ;

(2) Find the sum $T_{n}$ of the first $n$ terms of the sequence $\left\{ \frac{a_{n} + 1}{2^{n}} \right\}$ .