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If f ( x + y ) = f ( x ) f ( y ) and Σ x = 1 ∞ f x = 2 , x , y ∈ N , where N is the set of all natural numbers, then the value of f ( 4 ) f ( 2 ) is

Options

  1. A2 3
  2. B1 9
  3. C1 3
  4. D4 9

Correct answer

D. 4 9

Step-by-step solution

f ( x ) = a x satisfies the functional relation, so Σ x = 1 ∞ f x = 2 ⇒ a + a 2 + a 3 … … ∞   terms   = 2 ⇒ a 1 - a = 2 ⇒ a = 2 3 ⇒ f x = 2 3 x Now, f ( 4 ) f ( 2 ) = 2 3 4 2 3 2 = 4 9

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