p={1,10,100,1000,10000};

RecurrenceTable[{a[n + 1] == 2 a[n] + p[[n]], a[1] == 1}, a, {n, 1, 5}]

Part::pkspec1: The expression n cannot be used as a part specification.
Part::pkspec1: The expression #1 cannot be used as a part specification.

z = 1;
RecurrenceTable[{a[n + 1] == 2 a[n] + p[[z++]], a[1] == 1}, a, {n, 1, 5}]