geometric progression

a geometric progression is a sequence of the form
$\text{[math]}$ [cite:;taken from @discrete_kenneth_2018 chapter 2.4 sequences and summations; definition 2]

the formula for the $\text{[math]}$ th number in a geometric sequence with initial value $\text{[math]}$ and common ratio $\text{[math]}$ is: $\text{[math]}$

if $\text{[math]}$ and $\text{[math]}$ are real numbers and $\text{[math]}$ , then
$\text{[math]}$ [cite:;taken from @discrete_kenneth_2018 chapter 2.4.5 summations; theorem 1]

the sum of an infinite geometric sequence $\text{[math]}$ where $\text{[math]}$ is the first term and $\text{[math]}$ is the common ratio is
$\text{[math]}$