-3, 12, -48, ... Write the explicit and recursive formulas for the geometric sequence. Use your explicit formula to calculate the 6th term in the sequence.
looks like you are multiplying the current term by -4 to get the next term
Now translate "multiplying the current term by -4 to get the next term" into a recursive formula
for the explicit,
a = -3
r = -4
term(n) = a r^(n-1)
= -3(4)^(n-1)
term(6 ) = .....
the common ratio is -4
An = -3 * (-4)^(n - 1)
To find the explicit formula for a geometric sequence, we need to identify the common ratio (r) between consecutive terms.
To find the value of r, we can divide any term in the sequence by its previous term. Let's use the first two terms:
12 / (-3) = -4
Therefore, the common ratio (r) is -4.
The explicit formula for a geometric sequence is given by:
An = A1 * r^(n-1)
where An represents the nth term in the sequence, A1 is the first term, r is the common ratio, and n is the position of the term we want to find.
In this case, A1 = -3 and r = -4. Plugging these values into the explicit formula, we get:
An = -3 * (-4)^(n-1)
Now let's use this explicit formula to calculate the 6th term (n = 6):
A6 = -3 * (-4)^(6-1)
= -3 * (-4)^5
= -3 * (-1024)
= 3072
Therefore, the 6th term in the sequence is 3072.
The recursive formula for a geometric sequence is given by:
A(n) = A(n-1) * r
where A(n) represents the nth term in the sequence, A(n-1) is the previous term, and r is the common ratio.
In our case, the recursive formula would be:
A(n) = A(n-1) * (-4)
where A(n) represents the nth term and A(n-1) represents the previous term.
To find the explicit and recursive formulas for a geometric sequence, we need to identify the common ratio (r) between consecutive terms.
Looking at the given sequence: -3, 12, -48, ...
We can see that to go from -3 to 12, we need to multiply by -4, and to go from 12 to -48, we need to multiply by -4 again. So the common ratio (r) in this sequence is -4.
The explicit formula for a geometric sequence is given by:
an = a1 * r^(n-1)
Where:
an represents the nth term,
a1 represents the first term,
r represents the common ratio, and
n represents the position of the term in the sequence.
In this case, the first term (a1) is -3.
So, the explicit formula for this geometric sequence is:
an = -3 * (-4)^(n-1)
To find the 6th term using the explicit formula, we substitute n = 6:
a6 = -3 * (-4)^(6-1)
a6 = -3 * (-4)^5
Calculating this expression will give us the 6th term of the sequence.
Now, let's evaluate the 6th term using the explicit formula:
a6 = -3 * (-4)^5
a6 = -3 * (-1024)
a6 = 3072
Therefore, the 6th term in the sequence is 3072.
Now let's consider the recursive formula for the geometric sequence:
ar = ar-1 * r
Where:
ar represents the nth term, and
ar-1 represents the (n-1)th term.
In this case, the recursive formula would be:
a(n+1) = a(n) * (-4)
Where a(n) represents the nth term and a(n+1) represents the (n+1)th term.
I hope this helps! Let me know if you have any further questions.