Suppose f is a function satisfying f ( x + y ) = f ( x ) + f ( y ) for all x , y ∈ ℕ and f ( 1 ) = 1 5 . If ∑ n = 1 m f ( n ) n ( n + 1 ) ( n + 2 ) = 1 12 then m is equal to ______.

Question

Suppose f is a function satisfying f ( x + y ) = f ( x ) + f ( y ) for all x , y &#8712; &#8469; and f ( 1 ) = 1 5 . If &#8721; n = 1 m f ( n ) n ( n + 1 ) ( n + 2 ) = 1 12 then m is equal to ______.

Quantrex Academy · Accepted Answer

Correct answer: 0

Given, f x + y = f x + f y &#160; &#38; &#160; f 1 = 1 5 Now solving the given functional equation by taking x = 1 &#160; &#38; &#160; y = 1 we get, f 1 + 1 = f 1 + f 1 &#8658; f 2 = 2 5 &#160; as&#160; f 1 = 1 5 Now taking x = 2 &#160; &#38; &#160; y = 1 we get, f 3 = 2 5 + 1 5 &#8658; f 3 = 3 5 Hence, f x = x 5 Now solving &#8721; n = 1 m f n n n + 1 n + 2 = 1 12 &#8658; &#8721; n = 1 m n 5 n n + 1 n + 2 = 1 12 &#8658; 1 5 &#8721; n = 1 m 1 n + 1 n + 2 = 1 12 &#8658; 1 5 &#8721; n = 1 m 1 n + 1 - 1 n + 2 = 1 12 &

Suppose f is a function satisfying f ( x + y ) = f ( x ) + f ( y ) for all x , y ∈ ℕ and f ( 1 ) = 1 5 . If ∑ n = 1 m f ( n ) n ( n + 1 ) ( n + 2 ) = 1 12 then m is equal to ______.

Correct answer

Step-by-step solution

More from Continuity and Differentiability

Suppose f is a function satisfying f ( x + y ) = f ( x ) + f ( y ) for all x , y &#8712; &#8469; and f ( 1 ) = 1 5 . If &#8721; n = 1 m f ( n ) n ( n + 1 ) ( n + 2 ) = 1 12 then m is equal to ______.

Correct answer

Step-by-step solution

More from Continuity and Differentiability

Suppose f is a function satisfying f ( x + y ) = f ( x ) + f ( y ) for all x , y ∈ ℕ and f ( 1 ) = 1 5 . If ∑ n = 1 m f ( n ) n ( n + 1 ) ( n + 2 ) = 1 12 then m is equal to ______.