Geometric Progression: nth Term

A Geometric Progression (GP) is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed non-zero number. That fixed number is the common ratio, denoted $r$ (and no term is zero):

The first term is again $a$. Once $a$ and $r$ are known the whole GP is fixed: $a,\; ar,\; ar^2,\; ar^3, \ldots$

Example: for $2, 6, 18, 54, \ldots$ the first term is $a = 2$ and $r = \tfrac{6}{2} = 3$ (checked: $\tfrac{18}{6} = 3$, $\tfrac{54}{18} = 3$). The $6$th term is $t_6 = 2 \times 3^{5} = 2 \times 243 = 486$.