Example 5. Try to find the function $f(x)$ defined on the set of natural numbers, such that $f(x+y)=f(x)+f(y)+xy, f(1)=1$.

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Example 5. Try to find the function $f(x)$ defined on the set of natural numbers, such that $f(x+y)=f(x)+f(y)+xy, f(1)=1$.

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Please translate the text above into English, preserving the original text's line breaks and format, and output the translation directly.
Let $\mathrm{y}=1$, then we have
$$
f(x+1)=f(x)+x+1,
$$

i.e., $f(x+1)-f(x)=x+1$.
In the above equation, let $\mathrm{x}=1,2,3, \cdots, \mathrm{n}$ respectively, substitute them in, and add both sides of the equations respectively, considering $\mathrm{f}(1)=1$, we get $f(n)=\frac{1}{2} n(n+1)$.
This is the required functional relationship.