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, ...}
to find the common ratio, just divide any term by the one before it. Then you have the formula
A(n+1) = A(n)*r
Then you can use that to find the next two terms...
To find the recursive formula and the 5th term of the geometric sequence, we'll first determine the common ratio (r) of the sequence.
To find the common ratio (r), we can divide any term by the previous term. Let's divide the second term, 12, by the first term, -4.
r = 12 / -4
r = -3
The common ratio (r) is -3.
Now, let's find the recursive formula for the sequence. The recursive formula is a formula that expresses each term of a sequence as a function of the preceding term(s).
In this case, we can express each term in terms of the previous term by multiplying the previous term by the common ratio (-3).
The recursive formula for this geometric sequence is:
a(n) = -3 * a(n-1)
where a(n) represents the nth term of the sequence and a(n-1) represents the previous term in the sequence.
Now let's find the 5th term of the sequence using this recursive formula.
To find a(5), we can use the formula a(n) = -3 * a(n-1).
a(5) = -3 * a(5-1) (substituting n = 5)
a(5) = -3 * a(4)
To find a(4), we can use the same formula:
a(4) = -3 * a(4-1) (substituting n = 4)
a(4) = -3 * a(3)
Now let's find a(3) using the formula:
a(3) = -3 * a(3-1) (substituting n = 3)
a(3) = -3 * a(2)
Finally, let's find a(2):
a(2) = -3 * a(2-1) (substituting n = 2)
a(2) = -3 * a(1)
Now, we can substitute these values into the recursive formula to find a(5).
a(2) = -3 * a(1) = -3 * (-4) = 12
a(3) = -3 * a(2) = -3 * 12 = -36
a(4) = -3 * a(3) = -3 * (-36) = 108
a(5) = -3 * a(4) = -3 * 108 = -324
Therefore, the 5th term of the geometric sequence is -324.
To find the recursive formula for a geometric sequence, we need to determine the common ratio between consecutive terms.
The sequence given is {-4, 12, -36, ...}
To find the common ratio, we divide any term by its previous term. Let's take the second and first terms:
12 / (-4) = -3
Therefore, the common ratio is -3.
Now we can write the recursive formula for the sequence. Let's assume that the first term is denoted as a₁ and the common ratio as r.
a₁ = -4 (given)
The recursive formula for a geometric sequence is:
aₙ = r * aₙ₋₁
Substituting the given values, we have:
aₙ = -3 * aₙ₋₁
To find the 5th term in the sequence, we can use the recursive formula:
a₅ = -3 * a₄
To find a₄, we can use the recursive formula again:
a₄ = -3 * a₃
Substituting a₃:
a₄ = -3 * (-36) = 108
Now we can find the 5th term:
a₅ = -3 * 108 = -324
Therefore, the 5th term in the sequence is -324.