Final Answer:
The division method involves dividing two consecutive terms in the sequence or series to determine the common difference and then using that common difference to derive the explicit formula.
Step-by-step explanation:
The division method is a technique used to find the explicit formula for an arithmetic sequence or series. It is based on the fact that the common difference between any two consecutive terms in an arithmetic sequence is constant.
To use the division method, you first need to know two terms in the sequence or series. Let's call these terms a and b, where a is the first term and b is the second term.
Next, you need to divide a by b. If the result is an integer, then the sequence is arithmetic. If the result is not an integer, then the sequence is not arithmetic.
If the sequence is arithmetic, then the common difference is d = b - a.
Once you know the common difference, you can use the following formula to find the explicit formula for the sequence:
a_n = a + d(n - 1)
where:
a_n is the nth term in the sequence
a is the first term
d is the common difference
n is the term number
To find the explicit formula for an arithmetic series, you can use the following formula:
S_n = n/2 * (a + l)
where:
S_n is the sum of the first n terms in the series
a is the first term
l is the last term
n is the number of terms
Here is an example of how to use the division method to solve for the explicit formula for an arithmetic sequence:
Find the explicit formula for the sequence 3, 6, 9, 12, ...
First, divide the second term by the first term: 6 ÷ 3 = 2
Since the result is an integer, the sequence is arithmetic.
The common difference is d = 6 - 3 = 3
Therefore, the explicit formula for the sequence is:
a_n = 3 + 3(n - 1)