The given sequence is 12, 6, 3. To find the explicit formula for this geometric sequence, we first need to identify the common ratio between the terms. In a geometric sequence, each term is obtained by multiplying the previous term by a constant, known as the common ratio.
Let’s determine the common ratio (r):
- From the first term (12) to the second term (6):
- r = 6 / 12 = 0.5
- From the second term (6) to the third term (3):
- r = 3 / 6 = 0.5
Since the common ratio (r) is consistently 0.5 for the sequence, we can use this information to write the explicit formula for the geometric sequence.
The general formula for the nth term of a geometric sequence can be expressed as:
an = a1 * r(n-1)
Here:
- a1 = 12 (the first term of the sequence)
- r = 0.5 (the common ratio)
- n represents the term number
Substituting these values into the formula, we get:
an = 12 * (0.5)(n-1)
This explicit formula can be used to find any term in the geometric sequence. For example:
- To find the fourth term (n = 4):
- a4 = 12 * (0.5)(4-1) = 12 * (0.5)3 = 12 * 0.125 = 1.5
Therefore, the explicit formula for the geometric sequence 12, 6, 3 is:
an = 12 * (0.5)(n-1)