A Geometric Progression (G.P) is one in which the ratio of any term to its predecessor is always the same number. This ratio is called the common ratio. Thus, a sequence, a1, a2, ....., an is said to be in G.P.,
if \(\frac{a^2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=.....=\frac{a_n}{a_{n-1}}=r\)
where a1, a2, a3 ..... are all non-zero terms of the G.P. and r is the common ratio.
Examples of G.P.
| Progression |
Common ratio |
| (i) 1, 3, 9, 27, 81, .... |
r = 3 |
| (ii) \(\frac{16}{27}\), – \(\frac{8}{9}\), \(\frac{4}{3}\), – 2 .... |
r = – \(\frac{3}{2}\) |
| (iii) x, x2, x3, x4, .... |
r = \(x\) |
If a denotes the first term of a G.P. and r, the common ratio, then a standard G.P. is a, ar, ar2, ..... .
