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If g c d m , n = 1 and 1 2 - 2 2 + 3 2 - 4 2 + . . . . + 2021 2 - 2022 2 + 2023 2 = 1012 m 2 n then m 2 - n 2 is equal to

Options

  1. A240
  2. B200
  3. C220
  4. D180

Correct answer

A. 240

Step-by-step solution

Let S = 1 2 - 2 2 + 3 2 - 4 2 + . . . . + 2021 2 - 2022 2 + 2023 2 ⇒ S = 1 - 2 1 + 2 + 3 - 4 3 + 4 + . . . . + 2021 - 2022 2021 + 2022 + 2023 2 ⇒ S = - 3 + 7 + 11 + 15 + . . . . + 4043 + 2023 2 The number of terms in the bracket are 2022 2 = 1011 ⇒ S = - 1011 2 6 + 1010 × 4 + 2023 2 ⇒ S = - 1011 × 2023 + 2023 2 ⇒ S = 2023 × 1012 ⇒ S = 17 2 × 7 × 1012 So, m = 17 ,   n = 7 and g c d 17 , 7 = 1 Hence, m 2 - n 2 = 17 2 - 7 2 = 240 Hence this is the corre

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