7. Let the circle $C_{k}=\left\{(x, y) \mid\left(x-m_{k}\right)^{2}+(y-\right.$ $\left.\left.m_{k}\right)^{2} \leqslant 2 k^{2}\right\}\left(k \in \mathbf{N}_{+}\right)$, where $m_{k}$ is defined as follows: $m_{1}=0, m_{k+1}=m_{k}+2 k+1(k \geqslant 1)$.
Then the area of $\cup C_{k}$ is $\qquad$
$$
\text { II. 7. } \frac{\pi}{3} n(n+1)(2 n+1) \text {. }
$$

Using mathematical induction, it can be proven that $m_{k}=k^{2}-1$. It is easy to see that circle $C_{k}$ is externally tangent to circle $C_{k+1}$ $\left(k \in \mathbf{N}_{+}\right)$. Therefore, the area of $\bigcup_{1 \leqslant k \leqslant n} C_{k}$ is
$$
\sum_{k=1}^{n} \pi \cdot 2 k^{2}=\frac{\pi}{3} n(n+1)(2 n+1) .
$$