Question

It is given that $f(x)$ is a function defined on $N$, satisfying $f(1)=1$ and for any $x \in N$ $f(x+5) \geq f(x)+5 \quad \text { and } \quad f(x+1) \leq f(x)+1$ If $g(x)=f(x)+1-x$, then $g(2016)$ equals

Answer 1

$f(x)+5 \leq f(x+5) \leq f(x+4)+1 \leq f(x+3)+2 \leq f(x+2)+3 \leq f(x+1)+4 \leq f(x)+5$ $\Rightarrow \quad$ In all steps there is equality only $\quad \Rightarrow \quad f(x+1)=f(x)+1$

Now $f(1)=1$

$\Rightarrow \quad f(2)=2 \\$

$ f(3)=3 \\$

$ f(4)=4 \\$

$ f(2016)=2016 \\$

$\Rightarrow \quad g(2016)=2016+1-2016=1$