To find the explicit formula for the geometric sequence given by 12, 6, 3, we first identify the common ratio of the sequence. A geometric sequence is defined by each term being multiplied by the same factor, known as the common ratio.
Let’s calculate the common ratio (r):
r = second term / first term = 6 / 12 = 1/2
Now we will check the ratio with the next pair of terms:
r = third term / second term = 3 / 6 = 1/2
Both calculations give us the same ratio of 1/2, confirming that our sequence is geometric with a common ratio of 1/2.
Next, we need to construct the explicit formula for the sequence. The general form of a geometric sequence is:
an = a1 * r(n-1)
Where:
- an is the nth term of the sequence,
- a1 is the first term,
- r is the common ratio,
- n is the term number.
In our case:
- a1 = 12
- r = 1/2
Substituting these values into the formula provides us with:
an = 12 * (1/2)(n-1)
Thus, the explicit formula for the sequence 12, 6, 3 is:
an = 12 * (1/2)(n-1)