Let f : N → N be a function such that f m + n = f m + f n for every m , n ∈ N . If f 6 = 18 then f 2 · f 3 is equal to :

Step-by-step solution

Given functional equation is f m + n = f m + f n ; m , n ∈ N   . . . . 1 Put m = n = 3 f ( 3 + 3 ) = f ( 3 ) + f ( 3 ) ⇒ f ( 3 ) = 9 ∵ f 6 = 18 Put m = 2 , n = 1   in equation   1 ∴ f ( 3 ) = f ( 2 + 1 ) = f ( 2 ) + f ( 1 ) = f ( 1 + 1 ) + f ( 1 ) = f ( 1 ) + f ( 1 ) + f ( 1 ) 9 = 3 f ( 1 ) ⇒ f ( 1 ) = 3 ∴ f ( 2 ) = f ( 1 + 1 ) = f ( 1 ) + f ( 1 ) = 6 Now, f ( 2 ) · f ( 3 ) = ( 6 ) ( 9 ) = 54