Find function $F(s)$ such that $F(s+1) = F(s) - \frac{1}{(s-2)^2}$

Question

I want to find a function $F(s)$ such that $F(s+1) = F(s) - \frac{1}{(s-2)^2}$.
I first consider the function $\psi(s) = \frac{\Gamma'(s)}{\Gamma(s)}$. Then I have $\psi(s+1) = \frac{\Gamma'(s+1)}{\Gamma(s+1)}$. Then I use the identity that $\Gamma(s+1) = s\Gamma(s)$, and the product rule of the derivative. I get $$\psi(s+1) = \frac{\Gamma(s) - s\Gamma'(s)}{s\Gamma(s)}$$ Then I have $$\psi(s+1) - \psi(s) = \frac{\Gamma(s) + s\Gamma'(s) - s\Gamma'(s)}{s\Gamma(s)}$$ which is equal to $\frac{1}{s}$. But I'm confused about to get the desired function from here. Could anyone help me please? Thanks!
UPDATE I take the derivative on the both sides of the equation and I get $$\psi'(s+1) - \psi'(s) = -\frac{1}{s^2}$$ Then I think that my next step would be shifting $s$ to the right by 2
CONCLUSION I choose my $F(s)$ to be $\psi'(s-2)$. Then I have $$F(s+1) - F(s) = -\frac{1}{(s-2)^2}$$ which is desired