A series of numbers \(\left( {{t_n}} \right)\) is said to be in geometric progression (G.P.) when the first term is non zero and each successive term is r times the preceding term, i.e.\(\frac{{{t_n}}}{{{t_{n - 1}}}} = r\) is a constant. For e.g. the sequence 3, 9, 27, 81, ……. is in G.P., since,\(\frac{9}{3} = 3,\,\,\frac{{27}}{9} = \,3,\,\,\frac{{81}}{{27}} = 3\) and so on. The ratio of every term to its preceding term is a constant equal to 3.

In general, if the first term of a G.P. is \(a\) and the common ration is \(r\), then

\({t_1} = a,{t_2} = ar,{t_3} = a{r^2},{\rm{ }} \ldots  \ldots ,{t_n}\) = \(a{r^{n - 1}}\)

The sum of the first n terms of a G.P. is

\({S_n} = a + ar + a{r^2} + {\rm{ }} \ldots  \ldots  \ldots ..{\rm{ }} + \) \(a{r^{n - 1}}\)

\( = a\left( {\frac{{{r^n} - 1}}{{r - 1}}} \right)\,\,if\,r > 1\)

\( = a\left( {\frac{{1 - {r^n}}}{{1 - r}}} \right)\,\,if\,r < 1\)

If the GP is an infinite GP with common ratio 

\(r\) (where \(-1< r <1\)), then sum upto infinite terms is given by:  \(S = \frac{a}{{1 - r}}\)