Mathematica sometimes can't say much about "abstract" sums, where the upper bound is unknown.

This

x[1]=1;
Table[{t,(1-?)*Sum[?^(t-1)(P[t-k]-P[t-k-1]),{k,1,t-1}]},{t,2,6}]//Simplify

shows this

{{2, (?-1)*?  *(P[0] - P[1])},
 {3, (?-1)*?^2*(P[0] - P[2])},
 {4, (?-1)*?^3*(P[0] - P[3])},
 {5, (?-1)*?^4*(P[0] - P[4])},
 {6, (?-1)*?^5*(P[0] - P[5])}}

which implies this

x[t]==(?-1)?^(t-1)(P[0]-P[t-1])

which says this

P[t-1]== -x[t]/((?-1)?^(t-1))+P[0]

If you have some additional information about P[0] then this might help you write the definition of P[t]

Please check every step of this carefully