Question

Write the recursive rule for the nth term.

a) a_n = a_n-1 - 8
b) a_n = a_n-1 + 8
c) a_n = a_1 - 8(n-1)
d) a_n = a_1 + 8(n-1)

Answer 1

The student is seeking recursive rules for sequences' nth terms. Responses (a) and (b) already provide recursive rules indicating changes from one term to the next. Responses (c) and (d), however, are explicit formulas that compute the nth term directly from its position number.

Step-by-step explanation:

The student is asking for the recursive rule for the nth term of a sequence. In a recursive formula, each term is defined as a function of its preceding term(s). Here are the correct recursive rules for the sequences provided:

• (a) an = an-1 - 8: This is already in recursive form, indicating that each term is 8 less than the term before it.
• (b) an = an-1 + 8: Similar to part (a), but each term is 8 more than the term before it.
• (c) an = a1 - 8(n-1): This is an explicit formula, which directly calculates the nth term by using the first term and the position n in the sequence.
• (d) an = a1 + 8(n-1): Also an explicit formula, providing a method to find any term in the sequence without needing to know the previous term(s).

It's essential to note that parts (c) and (d) are not recursive rules but are instead explicit formulas. A recursive rule requires the previous term to find the current term, whereas an explicit formula provides a direct calculation for any term based on its position in the sequence.