Three. (Full marks 20 points) Let $a_{1}, a_{2}, \cdots$ and $b_{1}, b_{2}, \cdots$ be two arithmetic sequences, and their sums of the first $n$ terms are $A_{n}$ and $B_{n}$, respectively. It is known that for all $n \in N$, $\frac{A_{n}}{B_{n}}=\frac{2 n-1}{3 n+1}$. Try to write the expression for $\frac{a_{n}}{b_{n}}$ for all $n \in N$.
Three、 $\frac{4 n-3}{6 n-2}$