If you have difficulty memorizing the formula for geometric series and you'd rather not calculate the entire sum from 2 to 2^8, there's one other approach.

Back story:
Finding the sum 2 + 1 + 1/2 + 1/4 + 1/8 + . . . .+ 1/128 + 1/256 + 1/256 is difficult if you don't know the formula for the sum of a finite geometric series.
But, if we reword the question the answer to this related sum becomes clear.
Reworded question: I have 4 kg of gold. On the first day I give you half of my gold (2kg). On the second day, I give you half of what remains (1kg). On the third day, I give you half of what remains (1/2 kg) and so on.
As you can see, the amount of gold in your possession becomes the sum 2 + 1 + 1/2 + 1/4 + 1/8 + . . . .etc
Since I began with 4 kg of gold, this sum slowly gets closer and closer to 4 (kg)
At one point, I have 1/128 kg remaining. The next day, I give you half (1/256) and I give you the other half as well (1/256).
I have now given you all of my gold. (4kg)

Now apply this to the original question.
I have 2^9 kg of gold.
On day one, I give you half of that (2^8 kg).
On day two, I give you half of the remaining 2^8 kg (2^7 kg)
And so on.
You see that we get the same sum (2^8 + 2^7 + 2^6 + . . . +2+2)
Since we began with 2^9 kg of gold, the sum must be 2^9.