# Can this problem be generalised mathematically?

I found a mathematical riddle which I solved by experiment and I would like to know is there a formula of some sort to solve these kind of problems.

You have 10 Euro. You can buy a bottle of beer for 1 Euro. You can exchange 2 empty bottles for 1 new full bottle. What is the maximum number of bottles that you can go through for your 10 Euro ?

By experiment you can go through 19 bottles for the 10 Euro.

But is there some formula to obtain 19 from the initial conditions ? I thought maybe something related to sequences or some kind of limit.

Any ideas?

• "No single bottles remaining": does this mean that you end without drinking the last bottle? – abiessu Jul 19 '17 at 0:05
• if you had a geometric series, you would find an upper bound of 20 but that allows partial bottles. – user451844 Jul 19 '17 at 0:05
• because, you can't always hit half dead on unless you start with a power of two in this case. so in the geometric series you would get 10+5+2.5+1.25 and continue on forever to get 20 . in the series you want you would get 10+5+2+1+1 = 19 . – user451844 Jul 19 '17 at 0:25