By this Formula (a+1)^2 - a^2 = 2a + 1 = (a+1) + a

so

100^2-99^2 = 100+ 99

98^2 - 97 ^2 = 98+97

96^2-95^2 = 96 + 95

so on

adding we get 100^2-99^2+98^2-97^2+...+2^2-1^2 =100 + 99 + 98 ... + 1 = sum of integers from 1 to 100 = 100 *101/2 = 5050

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