are you sure thats the right formula? The denominator in your sum appears to be applying the same discount factor to all dividends, even though they happen at different times.
I think the formula you want is:
$\displaystyle \displaystyle P_0 = \left( \sum \frac{dps_t}{r_{div}^t} \right) + \frac{P_{t_x}}{r_{div}^{t_x}}$
whichever formula you use, provided $\displaystyle r_{div}$ is constant you can write $\displaystyle s=\frac{1}{r_{div}}$. You may be able to factorise depending on the values of $\displaystyle dps_t$ and $\displaystyle P_{t_x}$. Otherwise you have a cubic in s to solve, time to find a computer...
Sorry, you're right it should be $\displaystyle \displaystyle P_0 = \left( \sum \frac{dps_t}{r_{div}^t} \right) + \frac{P_{t_x}}{r_{div}^{t_x}}$
Where:
$\displaystyle {r_{div}} $ is constant;
the sum is maxed for 3 periods; and
x = 3
Hence $\displaystyle s=\frac{1}{r_{div}}$ won't work as the sum has only three periods.
I got to $\displaystyle {dps_1} - \frac{dps_3}{r_{div}^3} = ({P_0}-{P_3})(r-1)$ by eliminating $\displaystyle {dps_2}$ and $\displaystyle {dps_3}$.