Definition of GP

Geometric sequences

A sequence \(\{u_n\}\) which is such that

\[\frac{u_2}{u_1} = \frac{u_3}{u_2} = \dots = \frac{u_n}{u_{n-1}} = r,\]

where \(r\) is constant, is called a geometric progression (G.P.) and \(r\) is called the common ratio.

Denote the first term \(u_1\) by \(a\) (like in the case of an A.P., and so let us be careful about this), and standardly the common ratio by \(r\), the associate G.P. will have the first few terms written as

\[a, ar, ar^2, \dots ar^{n-1}, \dots\]

So, the \(n^{\text{th}}\) term of such an G.P. is given by

\[u_n = ar^{n-1}, \quad n = 1, 2, 3, \dots\]