$6 \cdot 29$ For any real number $x$, the function $f(x)$ has the property
$$
f(x)+f(x-1)=x^{2} \text {. }
$$

If $f(19)=94$, then what is the remainder when $f(94)$ is divided by 1000?
[Solution] From the property of $f(x)$, we get
$$
f(x)=x^{2}-f(x-1) \text {. }
$$

Repeatedly applying the above formula, we get
$$
\begin{aligned}
f(94)= & 94^{2}-f(93) \\
= & 94^{2}-93^{2}+f(92) \\
= & 94^{2}-93^{2}+92^{2}-f(91) \\
& \cdots \\
= & 94^{2}-93^{2}+92^{2}-91^{2}+\cdots+20^{2}-f(19) \\
= & 187+183+\cdots+43+400-94 \\
= & \frac{187+43}{2} \cdot\left(\frac{187-43}{4}+1\right)+400-94 \\
= & 4561 .
\end{aligned}
$$

Therefore, the remainder when $f(94)$ is divided by 1000 is 561.