Question

What is Geometric Sequences ?

Answer 1

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 \times 3=6`. In general, we have that `a_n=3a_{n-1}`.

If the starting point is `a`, and the ratio is `r`, we have that the generic element is given by `a_n=ar^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.