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, ...}
https://www.mathsisfun.com/algebra/sequences-sums-geometric.html
r - -3
a = -4
Xn = Xn-1 * -3 recursive
Xn = -4 (-3)^(n-1) general
= -4 , -4*-3, 12(-3) , -36 * -3
To find the recursive formula for a geometric sequence, we need to determine the common ratio (r) between consecutive terms. We can find this ratio by dividing any term by its previous term in the sequence.
Let's calculate the common ratio (r) for this sequence:
r = 12 / (-4) = -3
r = -36 / 12 = -3
As we can see, the common ratio (r) is -3 for this sequence.
Now that we have the common ratio, we can establish the recursive formula for a geometric sequence. The formula takes the form: an = r * an-1, where "a" represents the term index and "n" represents the term number.
So, in this case, the recursive formula for the given sequence is:
an = -3 * an-1
To find the 5th term in the sequence, we can use the recursive formula and work our way up from the initial term.
Let's go step by step:
a1 = -4 (Given)
a2 = -3 * a1 = -3 * (-4) = 12
a3 = -3 * a2 = -3 * 12 = -36
a4 = -3 * a3 = -3 * (-36) = 108
a5 = -3 * a4 = -3 * 108 = -324
Therefore, the 5th term in the sequence is -324.
To summarize:
Recursive formula: an = -3 * an-1
5th term of the sequence: -324
To find the recursive formula for a geometric sequence, we need to determine the common ratio.
Common ratio (r) can be found by dividing any term in the sequence by its preceding term.
Let's calculate the common ratio:
r = 12 / (-4) = -3
So, the common ratio (r) is -3.
The recursive formula for a geometric sequence is given by:
a(n) = a(n-1) * r
Where:
a(n) is the nth term in the sequence.
a(n-1) is the (n-1)th term in the sequence.
r is the common ratio.
Now, let's find the 5th term in the sequence using the recursive formula.
First, we know the first term of the sequence, a(1), is -4.
Now we can use the recursive formula to find subsequent terms:
a(2) = a(1) * r
= -4 * (-3)
= 12
a(3) = a(2) * r
= 12 * (-3)
= -36
a(4) = a(3) * r
= -36 * (-3)
= 108
a(5) = a(4) * r
= 108 * (-3)
= -324
Therefore, the recursive formula is a(n) = a(n-1) * (-3), and the 5th term in the sequence is -324.