10. (20 points) The sequence $\left\{a_{n}\right\}$ has the sum of the first $n$ terms $S_{n}$ satisfying
$$
a_{n}=\frac{1}{2}+\frac{1}{2} \sqrt{1+8 S_{n-1}} \quad (n \geqslant 2) \text {. }
$$

Also, $a_{1}=1$, find the general term formula of the sequence.