Question

Which equation represents the formula for the general term, gn, of the geometric sequence 3, 1, 1/3, 1/9, . . .?

Answer 1

Explicit form:
`g_n=3((1)/(3))^(n-1)`

Recursive form:
`g_n=(1)/(3)g_(n-1)` with
`g_1=3`.

Explanation:

The first term is 3 and we are dividing by 3 each time.

Another way to say we are dividing by 3 each time is to say we are multiplying by factors of 1/3.

If the first term is
`a` and
`r` is the common ratio then the geometric sequence in explicit form is:

`a_n=a(r)^(n-1)`

`g_n=3((1)/(3))^(n-1)`

The recursive form for a geometric sequence is
`a_n=ra_(n-1)` with
`a_1=a` where
`r` is the common ratio and
`a` is the first term.

So the recursive form for our sequence is
`g_n=(1)/(3)g_(n-1)` with
`g_1=3`.