# What is Geometric Sequences ?

Feb 1, 2015

A geometric sequence is given by a starting number, and a common ratio.

Each number of the sequence is given by multipling the previous one for the common ratio.

Let's say that your starting point is $2$, and the common ratio is $3$. This means that the first number of the sequence, ${a}_{0}$, is 2. The next one, ${a}_{1}$, will be $2 \setminus \times 3 = 6$. In general, we have that ${a}_{n} = 3 {a}_{n - 1}$.

If the starting point is $a$, and the ratio is $r$, we have that the generic element is given by ${a}_{n} = a {r}^{n}$. This means that we have several cases:

1. If $r = 1$, the sequence is constantly equal to $a$;
2. If $r = - 1$, the sequence is alternatively equal to $a$ and $- a$;
3. If $r > 1$, the sequence grows exponentially to infinity;
4. If $r < - 1$, the sequence grows to infinity, assuming alternatively positive and negative values;
5. If $- 1 < r < 1$, the sequence exponentially decreases to zero;
6. If $r = 0$, the sequence is constantly zero, from the second term on.