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Let S = (0, 2 ) : _ m=1 ⁶ ( + (m-1) 4 ) ( + m 4 ) = 4 2 . If and are the largest and smallest elements of S respectively, then the value of - is

Options

  1. A3
  2. B12
  3. C5 12
  4. D2

Correct answer

A. 3

Step-by-step solution

Let the given sum be E . We can write the product of cosecants as: E = _ m=1 ⁶ 1 ( + (m-1) 4 ) ( + m 4 ) Multiplying and dividing by ( 4 ) = 1 2 , we get: E = 2 _ m=1 ⁶ ( ( + m 4 ) - ( + (m-1) 4 ) ) ( + (m-1) 4 ) ( + m 4 ) E = 2 _ m=1 ⁶ [ ( + (m-1) 4 ) - ( + m 4 ) ] This is a telescoping sum. Expanding the terms: E = 2 [ ( - ( + 4 ) ) + ( ( + 4 ) - ( + 2 4 ) ) + + ( ( + 5 4 ) - ( + 6 4 ) ) ] E = 2 [ - ( + 3 2 ) ] Using the allied angle identity ( + 3 2 ) = - , we have: E = 2 ( + ) = 2 ( + ) = 2 = 2 2 (2 ) We are gi

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