Question

If $f(x+2)+f(x+5)=2015\forall$ x then period of the function $f\left(x\right)$ is

Answer 1

Let us put the values $x=0,1,-1,2,-2,3,-3$ to see the behaviour of the function

$x=0,f\left(2\right)+f\left(5\right)=2015f\left(2\right)+f\left(5\right)=f\left(5\right)+f\left(8\right)\Rightarrow f\left(2\right)=f\left(8\right)$

$x=1,f\left(3\right)+f\left(6\right)=2015f\left(3\right)+f\left(6\right)=f\left(0\right)+f\left(3\right)\Rightarrow f\left(0\right)=f\left(6\right)$

$x=-1,f\left(1\right)+f\left(4\right)=2015f\left(1\right)+f\left(4\right)=f\left(4\right)+f\left(7\right)\Rightarrow f\left(1\right)=f\left(7\right)$

$x=2,f\left(4\right)+f\left(7\right)=2015$

$x=-2,f\left(0\right)+f\left(3\right)=2015$

$x=0,f\left(5\right)+f\left(8\right)=2015$

this gives us the pattern that

$f\left(0\right)=f\left(16\right)$

$f\left(1\right)=f\left(7\right)$

$f\left(2\right)=f\left(8\right)$

$\Rightarrow f\left(x\right)=f(x+6)$

so the period of the function is $6$

to generalize it, $f(x+2)+f(x+5)=2015$ put $x=x-3$.

$f(x-1)+f(x+2)=2015$

$\Rightarrow f(x-1)=f(x+5)$ put $x=x+1$

$\Rightarrow f(x+(-1\left)\right)=f\left(x\right)=f(x+6)$