Maths discrete probability question

Hi,please could I have help on this question? I’m really confused on firstly, if p(x=r+1) 2/3(x=r), and to find x=1 we would have to use 0 which has a probability of 0, wouldn’t they all be 0?
Also, in the ms, why have they included p=6, wouldn’t that have a probability of 0?
Finally, what is the geometric progression for? Can’t we just use the sum is 1 to work out k?
Thank you.