S = 100 2 − 99 2 + 98 2 − 97 2 + . . . . . . . . . . . . . . . . . . . . . . . . + 2 2 − 1 2 ; what is the value of S? Options: A . &nbsp2020 B . &nbsp5050 C . &nbsp5024 D . &nbsp5000

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

Question

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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