Example 1. If for $m, n \in N, f(m+n)=\boldsymbol{f}(m) +f(n)+m n$, and $f(1)=1$, find the expression for $f(n)$.


Solve $f(m+1)=f(m)+m+1$.
Let $m=1,2, \cdots, n-1$, we have
$f(2)=f(1)+2$,
$$
f(3)=f(2)+3, \cdots, f(n)=f(n-1)+n .
$$

Adding both sides respectively, simplifying yields
$$
f(n)=f(1)+2+3+\cdots+n=\frac{1}{2} n(n+1) \text {. }
$$