Example 8 In the non-decreasing sequence of positive odd numbers $\{1,3,3,3,5,5,5,5,5, \cdots\}$, each positive odd number $k$ appears $k$ times. Find the general term $a_{n}$ of this sequence.
Solve: The subsequence generated by the sequence $\{1,3,3,3,5,5,5,5,5, \cdots\}$ is $(1),(3,3,3),(5,5,5,5,5)$, $\cdots,(2 k-1,2 k-1, \cdots, 2 k-1), \cdots$, where the $k$-th group consists of $2 k-1$ occurrences of $2 k-1$. If $a_{n}$ is in the $k$-th group, then $a_{n}=2 k-1$, and $1+3+7+\cdots+(2 k-3)+1 \leqslant n0$, solving this gives $\sqrt{n-1}<k \leqslant \sqrt{n-1}+1$.
Since $k \in \mathbf{Z}^{+}$, and $\sqrt{n-1}+1-\sqrt{n-1}=1$,
thus $k=[\sqrt{n-1}+1]=[\sqrt{n-1}]+1$, where $[x]$ denotes the greatest integer not exceeding $x$.
Therefore, $a_{n}=2 k-1=2[\sqrt{n-1}]+1$ is the desired result.