Does anyone know of an easy was to calculate the true EV of a bonus considering the house edge and playthrough requirement?

For example, lets say we receive a BTC1 bonus to play a game with a 10% HE and 10x playthrough.  

We can expect to lose BTC1 for every BTC10 wagered at this game, BTC10 is the amount we have to wager, but that doesn't make the EV of our bonus = 0

I think the calculation for the Value of the bonus looks something like this:

Starting BR = BTC1

BTC1 x .1 = .1
BTC1 - .1 =BTC0.9

Bonus Value with 1x Required = BTC.9

BTC0.9 x .1 = .09
BTC0.9 - .09 = BTC0.81

Bonus Value with 2x Required =  BTC0.81

BTC0.81 x .1 = .081
BTC0.81 - 0.81 = BTC0.729

Bonus Value with 3x Required = BTC0.729

Etc.

Am I doing it wrong?
If I'm doing it right, it there an easy way to calculate Bonus Value with 100x or more?

(Variance and deposit/no deposit are not relevant.)