In a betting game where the wager varies every bet - which wager should we use when the player wins

Question

In a betting game, let's say a slot machine, where the player places a different wager every time he bets, we want to have a bonus feature that pops up on the screen every 1 out 20 turns (on average) and gives the player x6 times on his bet. I am trying to show that giving the player x6 times on his latest wager (the one that he bet right before the bonus appeared) is the same as giving him x6 on his average wager until that bonus appeared (average of around 20 bets). I am trying to find the formal way to prove it, and also, I would like to know how it changes given different number of bets (the bonus is every 20 rounds, so after 200 tries he will get around 10 bonuses).