Suppose f is a function satisfying f(x+y)=f(x)+f(y) for all x, y ∈ N and f(1)=(1/5). If displaystyle∑n=1 text 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 ∈ N and f(1)=(1/5). If displaystyle∑n=1 text m (f(n)/n(n+1)(n+2))=(1/12), then m is equal to (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

∵ f(1)=(1/5) ∴ f(2)=f(1)+f(1)=(2/5) f(2)=(2/5) f(3)=f(2)+f(1)=(3/5) f(3)=(3/5) ∴ displaystyle∑n=1m (f(n)/n(n+1)(n+2)) =(1/5) displaystyle∑n=1m((1/n+1)-(1/n+2)) =(1/5)((1/2)-(1/3)+(1/3)-(1/4)+ ldots .+(1/m+1)-(1/m+2)) =(1/5)((1/2)-(1/m+2))=(m/10(m+2))=(1/12) ∴ m=10

Q. Suppose f is a function satisfying f(x+y)=f(x)+f(y) for all x,y∈N and f(1)=51​. If n=1∑m ​n(n+1)(n+2)f(n)​=121​, then m is equal to_____

∵f(1)=51​∴f(2)=f(1)+f(1)=52​ f(2)=52​f(3)=f(2)+f(1)=53​ f(3)=53​ ∴n=1∑m​n(n+1)(n+2)f(n)​ =51​n=1∑m​(n+11​−n+21​) =51​(21​−31​+31​−41​+….+m+11​−m+21​) =51​(21​−m+21​)=10(m+2)m​=121​ ∴m=10

Q. Suppose $f$ is a function satisfying $f (x + y) = f (x) + f (y)$ for all $x, y \in N$ and $f (1) = \frac{1}{5}$ . If $n = 1 \sum m \frac{f ( n )}{n ( n + 1 ) ( n + 2 )} = \frac{1}{12}$ , then $m$ is equal to_____