A series where the ratio of any term (except the first) to the previous term is constant is called a geometric series or geometric progression which we abbreviate as G.P. A G.P. may start with any number $a$. The ratio of any term to the preceding term is called the common ratio $r$. Thus $3 + 6 + 12 + 24 + 48$ is a G.P. with five terms, the first term is $a = 3$ and the common ratio is $r = 2$. In general for a G.P. with common ratio $r$ we have

the first term is $a_1 = a$

the second term is $a_2 = r \times a_1 = r\times a = ar^1$

the third term is $a_3 = r \times a_2 = r \times ar^1 = ar^2$

the fourth term is $a_4 = r \times a_3 = r \times ar^2 = ar^3$

et cetera and in general the $n$-th term is $a_n = ar^{n\mathbin{-}1}$

thus in general a G.P. of $n$ terms is represented as $$a + ar + ar^2 + ar^3 + \cdots + ar^{n\mathbin{-}1}$$

Note the last or $n$ th term is $ar^{n \mathbin{-}1}$ not $ar^n$. If $r$ is positive all terms are the same sign as $a$ but if $r$ is negative then the terms alternate in sing.

In the G.P $3 + 6 + 12 + 24 + 48$ the numbers 6, 12 and 24 are geometric means between 3 and 48. If $a, x, b$ form a G.P. then $x$ is the geometric mean between $a$ and $b$ and we have: $$\frac{b}{x} = \frac{x}{a}$$

$$x^2 = ab$$

$$x = \pm\sqrt{ab}$$

Thus the geometric mean is real only when $a$ and $b$ are of the same sign.

Exercises:

1. Find the G.P whose fifth term is 12 and whose 8th term is 324.

2. Find the G.P. with $a_1 + a_2 = 4$ and $a_2 + a_3 = 12$

3. Insert 3 geometric means between 2 and 162.

4. Calculate the tenth term of the G.P. with $a = 4$ and $r = 3$