I have come across this problem in lecture slides which i don't understand.
In a dice game there are 2 options: 1. play 2. stop
If we choose play, we get $4, and then roll the die. If the die rolls 1 or 2 we are forced to stop.
The slides show the expected earning (if you always choose play) from this game is $12. Specifically - I want to know how to compute the infinite recursive sum.
I think this is the model:
$$r_1 = (1/3)(4 * 1) + (2/3) (r_2)\\ r_2 = (1/3)(4 * 2) + (2/3) (r_3) \ldots\\ \ \\ E(r) = r_1 + r_2 + r_3 .... = 12$$
You are guaranteed $4$ from the first roll. The second is worth $\frac 23 \cdot 4$ because you have $\frac 23$ chance of doing it. The third is $(\frac 23)^24$ and the $k$th is worth $(\frac 23)^{k-1}4$. This gives you a geometric series. Do you know how to sum those?