Grand partition function (Volume divided into N spaces)

[anaisabel]

Homework Statement:

Homework: I have a model with volume V divided into N spaces, each space only has one atom. It is considered an ideal gas. I have to write the partition function and the average number of particles wich I did. But then I cant solve the next exercise(I will post picture of the equation I have to proof). the first picture is the calculations i did to obtain the partition function and the number of particles and the second picture is the equation i have to obtain, considering the grand partion function of the average number of particles.

Relevant Equations:

Grand partition function and average number of particles

equation i need to proof. the N in here, is the avarege number of particles, N0 is the total number of particles,V is total volume, v0 I am not quite sure what it is because it isn't mentioned in the homework, but I am assuming it is the volume of which space.