$100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2$

Let $N$ denote the number above. Find the remainder when $N$ is divided by $1000$ .

The answer is 100.

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Lets do pairing as follows:

100²-1²=101x99

99²-2²=101x97

-98²+3²= -101x95

......& so on to get 50 pairs

N=101x(99+97-95-93+91+89.......+3+1)

First alternate + & - cancels to give a result 4. This goes on till 48 terms so we get 24x4=96

We add the last +3+1 to get 100

So N=101x100=10100 & hence remainder 100 when divided by 1000