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AP EAMCET202023 Sep 2020Morning ShiftMathematicsFunctionsActual

Let R = ( 5 5 + 11 ) 2 n + 1 and f = R - R , where x denotes the greatest integer less than or equal to x , then R f =

Options

  1. A2 n + 1
  2. B2 2 n + 1
  3. C4 n + 1
  4. D4 2 n + 1

Correct answer

D. 4 2 n + 1

Step-by-step solution

Given, R = 5 5 + 11 2 n + 1 and f = R - R = R If I is the integral part of R , then R = I + f = 5 5 + 11 2 n + 1   … i and 0 < f < 1 Now since 5 5 - 11 = 0 . 18 < 1 Let us assume f 1 = 5 5 - 11 2 n + 1   … ii and 0 < f 1 < 1 equation i - equation ii , we get I + f - f 1 = 5 5 + 11 2 n + 1 - 5 5 - 11 2 n + 1 ⇒ I + f - f 1 = 2 C 1 2 n + 1 5 5 2 n 11 + C 3 2 n + 1 5 5 2 n - 2 11 3 + . . . . . . . . ⇒ I + f - f 1 = Even integer ⇒ f - f 1 must also be an intege

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