neeed help on this sequence problem.

Which one exactly are you stuck on?

Are we talking about that specific polygon, or a general polygon drawn on the grid?

general 1..

'I' conjecture...

Let p be the number of dots on the polygon and k the number of dots in the polygon.

Then the area of the polygon is given by p/2 + k - 1

Proving this may be a bit more difficult. Induction is probably the way forward.

EDIT: This is actually known as Pick's theorem.