# What is the common ratio of the geometric sequence 2, 6, 18, 54,...?

Feb 3, 2015

$3$

A geometric sequence has a common ratio, that is: the divider between any two nextdoor numbers:

You will see that $6 / 2 = 18 / 6 = 54 / 18 = 3$
Or in other words, we multiply by $3$ to get to the next.
$2 \cdot 3 = 6 \to 6 \cdot 3 = 18 \to 18 \cdot 3 = 54$

So we can predict that the next number will be $54 \cdot 3 = 162$

If we call the first number $a$ (in our case $2$) and the common ratio $r$ (in our case $3$) then we can predict any number of the sequence. Term 10 will be $2$ multiplied by $3$ 9 (10-1) times.

In general
The $n$th term will be$= a . {r}^{n - 1}$

Extra:
In most systems the 1st term is not counted in and called term-0.
The first 'real' term is the one after the first multiplication.

This changes the formula to ${T}_{n} = {a}_{0.} {r}^{n}$
(which is, in reality, the (n+1)th term).