Given the following geometric sequence, find the 12th term: {2, -4, 8, ...}.
-4096
2048
4096
-2048
(-2)^12 =?
not quite. The 12 term will be negative.
2(-2)^11 = -2^12
To find the 12th term of the given geometric sequence {2, -4, 8, ...}, we need to find the common ratio first.
To find the common ratio (r), we can divide any term by its previous term. Let's take the second and third terms:
8 / (-4) = -2.
Therefore, the common ratio (r) is -2.
To find the 12th term, we can use the formula for the nth term of a geometric sequence:
an = a1 * r^(n-1), where:
an = 12th term
a1 = first term
r = common ratio
n = term number
Substituting the values into the formula:
a12 = 2 * (-2)^(12-1)
a12 = 2 * (-2)^11
a12 = 2 * (-2048)
a12 = -4096
Therefore, the 12th term of the sequence {2, -4, 8, ...} is -4096.
To find the 12th term of a geometric sequence, you can use the formula for the nth term of a geometric sequence:
Terms(n) = a * r^(n-1)
Where:
- Terms(n) represents the nth term of the sequence.
- 'a' represents the first term of the sequence.
- 'r' represents the common ratio between consecutive terms.
- 'n' represents the position of the term we want to find.
In this case, we're given the geometric sequence {2, -4, 8, ...}, where the first term (a) is 2. To find the common ratio (r), we can divide any term by the preceding term:
Common ratio (r) = -4 / 2 = -2
Now that we have the first term (a = 2) and the common ratio (r = -2), we can substitute these values into the formula:
Terms(n) = 2 * (-2)^(n-1)
We want to find the 12th term, so we substitute n = 12 into the formula:
Terms(12) = 2 * (-2)^(12-1)
= 2 * (-2)^11
= 2 * (-2048)
= -4096
Therefore, the 12th term of the given geometric sequence is -4096.