determine the number of terms for this geometric sequence?

We need to solve this :

(p^2 / 2 ) * (p/2)^(n -1)  = p^9 / 256    where n is the nth term

Multiply through by  2 / p^2

(p / 2)^(n -1)  =  p^7 / 128

Note   p^7 / 128  =   p^7 / 2^7  =  ( p / 2)^7

So......

(p / 2) ^(n - 1)  =  (p /2 )^7

Since the bases are the same, solve for the exponents

n - 1  = 7      add 1 to both sides

n  =  8   terms