Question

For the following geometric sequence find the recursive formula and the 5th term in the sequence. In your final answer, include all of your work.
{-4, 12, -36, ...}

Answer 1

We have a geometric sequence----------- > {-4, 12, -36, ...}

the formula is a(r)^(n-1)

a------------- >a is the first term------------ > -4
r--------------- > is the common ratio------- > 12/(-4)=(-36/12)=-3
n--------------- > is the number of terms

The fifth term is -4[(-3)^(5-1)]=-4[(81]=-324

the answer is -324

Answer 2

First you need to identify the common ratio:

r = $t_{n+1} / t_{n}$

where r is the common ratio
          $t_{n}$ is any term in the sequence
          $t_{n}$ is the term preceding $t_{n}$

In your case the common ratio will be 
  12/-4 = -3
The recursive formula of a geometric sequence is:

An = $A_{n-1}$ x r
Where : An is the nth term
             $A_{n-1}$ is the erm preceding the nth term
             r = common ratio

For this case it is:
An = $A_{n-1}$ x -3

Now let us use this formula to find the 5th term:

$A_{5}$ = $A_{5-1}$ x -3
$A_{5}$ = $A_{4}$ x -3

Since you do not know the 4th term, you can use that by using our reclusive formula:

$A_{4}$ = $A_{4-1}$ x -3
$A_{4}$ = $A_{3}$ x -3
$A_{4}$ = $-36$ x -3
$A_{4}$ = $108$

Now that you know your fourth term, you can use the same formula:

$A_{5}$ = $A_{5-1}$ x -3
$A_{5}$ = $A_{4}$ x -3
$A_{5}$ = $108$ x -3
$A_{5}$ = $-324$