Finding Price Paths from Difference Equations

$\displaystyle Qt^{s}$ = Quantity supplied at time t

$\displaystyle Qt^{d}$ = Quantity demanded at time t

$\displaystyle Pt$ = Price of good at time t

$\displaystyle Qt^{s}$ = -Pt +2Pt-1

$\displaystyle Qt^{d}$ = 50 – 0.5Pt

The market is in equilibrium $\displaystyle Qt^{d} = Qt^{s}$

I am asked to find the price path for Pt if P0 = 30

Im a little unsure as to whether I’ve plugged the right info into the formula

$\displaystyle 50-0.5Pt = -Pt + 2Pt-1$

Rearranging that I get

$\displaystyle Pt=A(\frac {-2}{0.5})$$\displaystyle ^{t} + \frac{1+50}{0.5 + 2}$

$\displaystyle = A(\frac{-2}{0.5})^{t}+ 20.4$

$\displaystyle A = [30 - (\frac{1+50}{0.5+2})$

$\displaystyle =[30-20.4] = 9.6$

Therefore I get

$\displaystyle Pt=9.6(\frac{-2}{0.5})^{t} + 20.4 $

I’m unsure as to whether I’ve actually done this correctly, as my understanding on this isn’t that great, any help would be very much appreciated.