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, ...}
12/-4 = -3
-26/12 = -3
so, clearly
An+1 = An * -3
Now you know you have
a = -4
r = -3
and use your formula to get the 5th term.
To find the recursive formula and the 5th term in the given geometric sequence {-4, 12, -36, ...}, we need to identify the ratio between consecutive terms.
The ratio can be found by dividing any term in the sequence by its preceding term. Let's choose the second term (12) and divided it by the first term (-4):
Ratio = 12 / -4 = -3
The ratio between consecutive terms is -3.
Now, let's write the recursive formula for this sequence. A recursive formula expresses each term in the sequence as a function of the preceding terms.
Using the ratio we found, the recursive formula will be:
a(n) = r * a(n-1)
Where:
a(n) represents the nth term in the sequence,
r represents the common ratio, and
a(n-1) represents the previous term in the sequence.
Applying this to the given sequence, we have:
a(n) = -3 * a(n-1)
Next, we need to find the 5th term in the sequence.
To do this, we can use the recursive formula by replacing n with the value 5 and using the fourth term (-36) as a starting point.
a(5) = -3 * a(5-1)
a(5) = -3 * a(4)
Now, we substitute a(4) with the third term (-36):
a(5) = -3 * (-36)
a(5) = 108
The 5th term of the sequence is 108.
Therefore, the recursive formula for the given geometric sequence is a(n) = -3 * a(n-1), and the 5th term of the sequence is 108.