Final answer:
Yes, with the seventh term and common difference of an arithmetic sequence, you can write a recursive formula. Calculate the first term using the seventh term and common difference, then apply the recursive formula where each term is the sum of the previous term and the common difference.
Step-by-step explanation:
If you know the seventh term in an arithmetic sequence and the common difference, you can indeed write a recursive formula for the sequence. A recursive formula defines each term of a sequence using the previous terms. In the case of an arithmetic sequence, each term is found by adding the common difference to the previous term. To write the recursive formula, you need the first term, which can be calculated if you know the seventh term and the common difference.
The recursive formula of an arithmetic sequence is generally expressed as:
an = an-1 + d,
where an is the nth term, an-1 is the previous term, and d is the common difference.
Let's assume the seventh term is a7 and the common difference is d. First, find the first term (a1) by using the formula:
a7 = a1 + 6d.
Now, solve for a1 to find its value:
a1 = a7 - 6d.
After finding the first term, the recursive formula can be written as:
an = an-1 + d where a1 is the calculated first term.
It is important to check the answer to see if it is reasonable to confirm that you have applied the formula correctly.