S=1002−992+982−972+........................+22−12; what i

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$S = 100^{2} - 99^{2} + 98^{2} - 97^{2} + . . . . . . . . . . . . . . . . . . . . . . . . + 2^{2} - 1^{2};$ what is the value of S?

Options:

A . 2020

B . 5050

C . 5024

D . 5000

Answer: Option B
:
B

$(100^{2} - 99^{2}) + (98^{2} - 97^{2}) + . . . . . . . . . . . . . . . . . . + (2^{2} - 1^{2})$

= 199 + 195 + 191 + 187 + ......................... + 3
$= (199 + 3) \times \frac{50}{2} = 5050.$
Even simpler way of calculation would be:
(100+99)+(98+97)+...+(2+1)
= Sum of first 100 natural numbers $= \frac{100 \times 101}{2} = 5050$

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