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The number of functions f : 1 , 2 , 3 , 4 → a ∈ ℤ : | a | ≤ 8 satisfying f ( n ) + 1 n f ( n + 1 ) = 1 , ∀ n ∈ 1 , 2 , 3 is

Options

  1. A3
  2. B4
  3. C1
  4. D2

Correct answer

D. 2

Step-by-step solution

Given, f ( n ) + 1 n f ( n + 1 ) = 1 , ∀   n ∈ 1 , 2 , 3 ⇒ n f ( n ) + f ( n + 1 ) = n At n = 1 , f ( 1 ) + f ( 2 ) = 1 ....(1) At n = 2 , 2 f ( 2 ) + f ( 3 ) = 2 ....(2) At n = 3 , 3 f ( 3 ) + f ( 4 ) = 3 ....(3) Put the value of f ( 2 ) from equation (1) in equation (2), 2 ( 1 - f ( 1 ) ) + f ( 3 ) = 2 ⇒ f ( 3 ) = 2 f ( 1 ) ....(4) Put the value of f ( 3 ) from equation (4) in equation (3), 3 ( 2 f ( 1 ) ) + f ( 4 ) = 3 ⇒ f ( 4 ) = 3 - 6 f ( 1 ) ∵   f : 1 , 2 , 3 , 4

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