find period of function

let the function be
f(x-1) + f(x+1) = root 3 f(x)
putting x=x+1
it can be written as..
f(x) + f(x+2) = root3f(x+1)...........(1)

now putting x=x+2
we get

f(x+2) + f(x+4) = root3f(x+3)....................(2)

adding (1) and (2)]

f(x) + 2f(x+2) + f(x+4) =root3(f(x+1) + f(x+3))=root3(root3f(x+2))=3f(x+2)

so f(x) + f(x+4) = f(x+2)

adding both we get....
f(x) + f(x+6) =0,,,,,,,,,,,,,,,so f(x) = -f(x+6)

so now f(x+12 ) = -f(x+6) =f(x)

so the period is 12