Question

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

Answer 1

Answer:

Explicit form: $g_n=3(\frac{1}{3})^{n-1}$

Recursive form: $g_n=\frac{1}{3}g_{n-1}$ with $g_1=3$.

Step-by-step 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:

$a_n=a(r)^{n-1}$

$g_n=3(\frac{1}{3})^{n-1}$

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 $g_n=\frac{1}{3}g_{n-1}$ with $g_1=3$.