# (why) is this ratio the golden ratio?

Looking at a slight variation of the fibonacci sequence
f(x) = f(x-1) + f(x-2) + 1
where f(1) = 1, f(2) = 1

I'm trying to find the ratio of this sequence but can't figure out how. To get an approximation I just tried looking at some random examples, (i.e. f(10)/f(9), f(20)/f(19))
And it seems that this is the golden ratio, but I can't understand how that is correct as this sequence seems to grow much faster than the fibonacci sequence

• We can tell that if this modification of the Fibonacci sequence tends to a geometric sequence (as the Fibonacci sequence itself does), then the common ratio must also be $\varphi$, since if we divide by $f(x-2)$ and take the limit, we get $r^2 = r + 1$ (since $1/f(x-2) \to 0$), which is the same characteristic equation as the Fibonacci series has. Apr 23 '20 at 20:15

If you write $$g(x)= f(x)+1$$ you got $$g(x+1)=g(x)+g(x-1)$$