Geometric sequences | Mathematico

Here is an example of what we call a geometric sequence:

3, 6, 12, 24, 48, 96, \dots

As you can see, to get the next term we just multiply by $2.$ For this sequence, $2$ is called the common ratio.

More generally, a geometric sequence is a sequence of the form

a, a r, a r^{2}, a r^{3}, \dots

where $a$ is the first term and $r$ is the common ratio.

We can see that

\begin{aligned} u_{1} & = a \\ u_{2} & = a r \\ u_{3} & = a r^{2} \\ u_{4} & = a r^{3} \end{aligned}

so that the $n^{th}$ term is given by the explicit formula

u_{n} = a r^{n - 1}

or the recursive formula

\begin{aligned} u_{1} & = a \\ u_{n + 1} & = u_{n} r \end{aligned}

A geometric sequence begins

2, \frac{14}{3}, \frac{98}{9}, \dots