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JEE MainMathematicsInverse Trigonometric Functions

The value of the sum _ n=1 ¹⁰ ⁻¹( n) is equal to

Options

  1. A14 - 43
  2. B4 + 5
  3. C22 - 43
  4. D55

Correct answer

B. 4 + 5

Step-by-step solution

The function f(x) = ⁻¹( x) is periodic with period 2 . Its principal value branch is [0, ] . For n=1, 2, 3 , n [0, ] (since 3.14 ). ⁻¹( n) = n Sum for n=1, 2, 3 is 1 + 2 + 3 = 6 . For n=4, 5, 6 , n [ , 2 ] (since 2 6.28 ). ⁻¹( n) = 2 - n Sum for n=4, 5, 6 is (2 - 4) + (2 - 5) + (2 - 6) = 6 - 15 . For n=7, 8, 9 , n [2 , 3 ] (since 3 9.42 ). ⁻¹( n) = n - 2 Sum for n=7, 8, 9 is (7 - 2 ) + (8 - 2 ) + (9 - 2 ) = 24 - 6 . For n=10 , n [3 , 4 ] (since 4 12.57 ). ⁻¹( 10) = 4 - 10 Adding all these sums together: Total sum =

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