Spider's square nest

Consider constructing the $n^{th}$ grid from the $(n-1)^{th}$ grid.

If we construct the top left and bottom right corner squares first, we have used $6$ Matchsticks.

Then, if we construct the top row one square at a time (from the top left corner moving right), apart from the top right corner square, we will require $2$ Matchsticks for each of the $(n-2)$ squares, meaning we will need $(2n -4)$ Matchsticks.

Similarly, if we construct the right column one square at a time (from the bottom right corner moving up), apart from the top right corner square, we will require $(2n -4)$ Matchsticks.

Finally, the top right corner square will require $2$ Matchsticks to finish the $n^{th}$ grid.

This means that to construct the $n^{th}$ grid from the $(n-1)^{th}$ grid, we need $6+2(2n-4)+2 = 4n$ Matchsticks.

In the case of the $1000^{th}$ grid, we require $4 * 1000 = 4000$ Matchsticks.