What is the geometric sequence for `3, 6, 12, 24, ...`?

In a geometric sequence, the terms are separated by a common ratio `r`. So, for example, the 4th term `a_4` will be `rxx a_3`, the 3rd term `a_3=r xx a_2`, and so on. From this we can get a general formula for the `n^"th"` term in terms of `r` and the first term `a_1`:

`a_n= r xx a_(n-1)"                  "=r^1a_(n-1)`
`color(white)(a_n)=r xx(r xx a_(n-2))"        "=r^2a_(n-2)`
`color(white)(a_n)=r xx r xx (r xx a_(n-3))" "=r^3a_(n-3)`
`color(white)(a_n)=...`
`a_n=r^(n-1)xxa_(n-(n"-"1))"     "=r^(n-1)a_1`

This is often written with the initial value `a_1` (often just called `a`) in front, like this:

`a_n=ar^(n-1)`

For the sequence `3, 6, 12, 24`, we are given the first term `a=3`. Now all we need is the common ratio `r`. This can be found by computing the ratio of any two successive terms.

(That is, since `a_2=r xx a_1` for any geometric sequence, we can find `r` by solving `r=a_2/a_1`, or `r=a_3/a_2`, or in general `r=a_k/a_(k-1)`.)

Using `a_2=6` and `a_1=3`, we get

`r=a_2/a_1=6/3=2`.

Thus, the common ratio is 2, the first term is 3, and so the formula for this geometric sequence is

`a_n=ar^(n-1)`
`color(white)(a_n)=3*2^(n-1)`.