Task 1. Determine the value of the numerical expression

$$
8004^{2}-8003^{2}+8002^{2}-8001^{2}+\ldots+4^{2}-3^{2}+2^{2}-1^{2}
$$
Solution.

$$
\begin{aligned}
& 8004^{2}-8003^{2}+8002^{2}-8001^{2}+\ldots+4^{2}-3^{2}+2^{2}-1^{2}= \\
&=(8004-8003)(8004+8003)+(8002-8001)(8002+8001)+\ldots \\
&+(4-3)(4+3)+(2-1)(2+1) \\
&= 8004+8003+8002+8001+\ldots+4+3+2+1 \\
&=(8004+1)+(8003+2)+\ldots+(4003+4002) \\
&= 4002 \cdot 8005=32036010
\end{aligned}
$$