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, ...}

notice that each new term is -3 times its previous, or

tn = -3t(n-1)

so we have -4, 12, -36, 108, -324

To find the recursive formula and the 5th term of a geometric sequence, we need to identify the common ratio (r) between consecutive terms. We can do this by dividing any term by its preceding term.

Let's start by dividing the second term (12) by the first term (-4):
12 / (-4) = -3

Now, let's divide the third term (-36) by the second term (12):
-36 / 12 = -3

We can conclude that the common ratio (r) is -3 since it is consistent for each division.

Now that we know the common ratio (-3), we can write the recursive formula for a geometric sequence. The formula is often denoted as:
aₙ = r * aₙ₋₁

Here, aₙ represents the nth term in the sequence, aₙ₋₁ represents the (n-1)th term, and r is the common ratio.

Since we know the first term (a₁) is -4, we can find the fifth term (a₅) by applying the recursive formula four times, as the fifth term follows the pattern of the previous term:
a₂ = -3 * (-4) = 12
a₃ = -3 * 12 = -36
a₄ = -3 * (-36) = 108
a₅ = -3 * 108 = -324

The recursive formula for the geometric sequence {-4, 12, -36, ...} is aₙ = -3 * aₙ₋₁, and the fifth term (a₅) is -324.