If f(x+1/2) + f(x-1/2) = f(x) for all x belonging to real numbers. Then the period of f(x) is equal to??

let x--->x+1/2

then f(x+1) + f(x) = f(x+1/2)

but f(x+1/2) + f(x-1/2) = f(x) then f(x+1) = -f(x-1/2)

now let x----->x+1/2

so, -f(x+3/2) = f(x)

and from this equation we get f(x+3/2) = -f(x+3)

so we get f(x)=f(x+3)

so the period is 3.

Approved