So i am working through a proof in a paper and so far have managed to obtain the expressions
$$f(x) = \frac{\theta\Gamma(J+1)\Gamma(J+\theta-x)}{x\Gamma(J+\theta)\Gamma(J+1-x)} \quad(1) $$
$$r = \sum_{x=1}^{J}f(x) \quad(2)$$
Now in the paper i'm working through it somehow uses the fact that $$ \sum_{x=1}^Jxf(x) = J \quad(3)$$ to obtain recurrence relationship
$$r(J+1) = r(J) + \frac{\theta}{\theta+J} (4) $$ And finished by using the fact that $r(1) = 1$ to find $$r = \sum_{i=0}^{J-1}\frac{\theta}{\theta+i} \quad (5)$$
Now i understand how they got (1) (2) and (3) and i also understand how to get from (4) to (5). However i have no clue how they get to (4)
If it helps here is a link to the paper or the preprint. The specific equation i'm trying to get to is (7) though most of the relevant information is in the Appendix.