Question

Does a recursive rule for an arithmetic sequence always need to include at least one term of the sequence? Explain.

Answer 1

Yes, a recursive rule for an arithmetic sequence always needs to include at least one term of the sequence because the recursive rule defines the relationship between terms. Without a known term to start with, it is impossible to determine the subsequent terms using the rule.

Step-by-step explanation:

Yes, a recursive rule for an arithmetic sequence always needs to include at least one term of the sequence. This is because a recursive rule defines how each term in the sequence is related to the previous term(s). In order to determine the value of a term using the recursive rule, you need a known term to start with.

For example, let's consider an arithmetic sequence with a first term of 3 and a common difference of 2:

1. The first term is 3.
2. The recursive rule states that to find the next term, you add 2 to the previous term: an = an-1 + 2.
3. To find the second term, you plug in the first term into the recursive rule: a2 = a1 + 2 = 3 + 2 = 5.

Without knowing the first term, it would be impossible to use the recursive rule and determine the value of subsequent terms in the sequence.