is it possible to solve it for variable r?

Question

Here is the annuitet payment formula: $$p = s \cdot \left(r + \frac{r}{(r+1)^t-1}\right)$$

Is it possible to solve it for rate ?

Answer 1

You have $$ \begin{split} \frac{p}{s} = \frac{r(r+1)^t}{(r+1)^t-1} = \frac{r}{1-(r+1)^{-t}} \end{split} $$ so for general real $t$, no analytic solution would exist, this is a transcendental equation (kind of like solving $xe^x = c$).

But the RHS is a nice function for $r > 0$, you can certainly solve the equation numerically using a root-finder, like Newton's or Bisection methods.