Three, (16 points) Given that $n$ is a positive integer, the graph of the linear function $y=$ $\frac{n+1}{n} x+n+1$ forms a triangle with the coordinate axes. The area of the circumcircle of this triangle is $\frac{25}{4} \pi$. Find the equation of this linear function.

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The graph of the linear function $y=$ $\frac{n+1}{n} x+n+1$ intersects the coordinate axes to form a triangle. The area of the circumcircle of this triangle is $\frac{25}{4} \pi$. Find the equation of this linear function.
Three, as shown in Figure 12, we easily get
$$
A(-n, 0), B(0, n+1) \text {. }
$$

From the given information,
$$
\left(\frac{|A B|}{2}\right)^{2} \pi=\frac{25}{4} \pi \text {, }
$$

we get $|A B|=5$.
By $|A O|^{2}+|B O|^{2}=|A B|^{2}$, we have
$$
n^{2}+(n+1)^{2}=25 \text {. }
$$

Thus, $n=3$.
Therefore, the equation of the linear function is $y=\frac{4}{3} x+4$.