A geometric sequence is built with an initial value and a common ratio.
You start with the initial value [tex] x_0 [/tex], which is the first element of the sequence, and then you build the term [tex] x_i [/tex] by multiplying the previous element [tex] x_{i-1} [/tex] by the common ratio [tex] r [/tex]. So, the first terms of the sequence are
[tex] x_0,\ x_1 = x_0\cdot r,\ x_2 = x_0\cdot r^2,\ x_3 = x_0\cdot r^3, \ldots [/tex]
In this sequence, every element is equal to the previous one, except the sign is opposite. This means that each element is computed by multiplying the previous one by [tex] -1 [/tex], which is thus the common ratio of the sequence.
