Problem: The function $f_{}^{}$ has the property that, for each real number $x,\,$ $f(x)+f(x-1) = x^2\,$. If $f(19)=94,\,$ what is the remainder when $f(94)\,$ is divided by $1000$? Solution: We can rewrite to get $f(x) = x^2-f(x-1)$. In sum, we get $\begin{aligned} f(94) & =\left(\sum_{n=20}^{94}(-1)^nn^2\right) - f(19)\\ & =\left(94^2-93^2\right)+\left(92^2-91^2\right)+\cdots+\left(22^2-21^2\right)+20^2-f(19) \\ & =94+93+\cdots+21+400-94 \\ & =4561\end{aligned}$