Ok, so it's all differences of squares, which will factor nicely. (1-2)(1+2)+(3-4)(3+4)..... The first factor will always be -1, so we can factor it out. (-1)*[ ( 1+2) + (3+4) + ...(198+199) ]. So we have just the negative of the sum of integers from 1 to 199.
Using Gauss' childhood trick, we have 100 (1+199) so a final result of -20000. I check this idea for 1-4+9-16 and it gives -10 which makes sense. Feeling self-satisfied but not 100% confident, I punch it into Desmos and get 19900. Ach! I've missed something!
I see it! My expression ends with a negative 199^2, but the series ends with a positive. I'll trim off the 199, find the sum, and then add it back. The sum should be (198/2)*(1+198)=19701 plus 199 gives me 19900, which I know should be -19900. A nice problem over Sunday breakfast!
Good work, but you retained a negative somewhere along the way... I believe the answer is +19900.
Yes! Funny, that. I found the correct answer, but poisoned it by pulling back in an earlier thought that it would be negative because you are always subtracting a larger number. I didn't reconsider this when I removed the 199^2.
Remove the 1^2 instead. Now it's all positive. Does that make it easier?
Sure. I thought of that later. 😀