Geometric Progression (G. P.)

Let us consider the following sequences:  2,4,8,16,..., 
we have `a_1 = 2 , a_2/a_1 = 2 , a_3/a_2 = 2, a_4/a_3 = 2` and so on. 
In above sequence the constant ratio is 2.  Such sequences are called geometric sequence or geometric progression abbreviated as G.P.

1) General term of a G .P:
Let us consider a G.P. with first non-zero term ‘a’ and common ratio ‘r’. The second  term is obtained by multiplying a by r, thus `a_2` = ar. Similarly, third term is obtained by multiplying `a_2` by r. Thus, `a_3 = a_2r = ar^2`, and so on.
The nth term of a G.P. is given by `a_n =ar^(n-1)`.
The series `a + ar + ar^2 + ... +` `ar^(n–1)`or `a + ar + ar^2 + ...+` `ar^(n–1) +...` are called finite or infinite geometric series, respectively.

2)  Sum to n terms of a G .P.:
 The first term of a G.P. be a and the common ratio be r.
`s_n =(a(1-r^n))/1-r`        or  `s_n = (a(r^n -1))/r-1`

3)  Geometric Mean (G .M.):
The geometric mean of two positive numbers a
and b is the number `sqrt (ab)` .
`G_1, G_2,…, G_n` be  n numbers between positive numbers a and b such that a,`G_1,G_2,G_3,…,G_n`, b is a G.P.

`G_n =ar^n =a (b/a)^(n/(n + 1))`