12. Given the following conditions for the sequence $\left\{a_{n}\right\}$: (1) $\left\{2^{a_{n}}\right\}$ is a geometric sequence; (2) the sum of the first $n$ terms $S_{n}=n^{2}+1$; (3) $a_{1}>0$, and $a_{k}=\frac{2}{k-1}\left(a_{1}+a_{2}+\cdots+a_{k-1}\right)(k \geqslant 2)$. The condition that makes $\left\{a_{n}\right\}$ an arithmetic sequence is $\qquad$ (fill in the sequence number).
12. (1) (2).
$2\left(a_{1}+a_{2}+\cdots+a_{k-1}\right)=(k-1) a_{k}, 2\left(a_{1}+a_{2}+\cdots+a_{k}\right)=k a_{k+1}(k \geqslant 2)$, subtracting gives $a_{k+1}=$ $\frac{k+1}{k} a_{k}$, thus $a_{k}=\frac{k}{k-1} a_{k-1}=\frac{k}{k-1} \cdot \frac{k-1}{k-2} a_{k-2}=\cdots=k a_{1}$.