Question

Shaunta is developing a recursive formula to represent an arithmetic sequence in which 5 is added to each term to determine each successive term. Which formula could represent her sequence?

f(n + 1) = f(n) + 5
f(n + 1) = f(n + 5)
f(n + 1) = 5f(n)
f(n + 1) = f(5n)

Answer 1

To develop a recursive formula for an arithmetic sequence where 5 is added to each term to determine each successive term, you want to express the next term in terms of the current term plus 5.

The general form of a recursive formula for such an arithmetic sequence is:
\[ f(n + 1) = f(n) + d \]
where \( d \) is the common difference between terms.

In this case, the common difference (\( d \)) is 5. Therefore, the appropriate recursive formula is:
\[ f(n + 1) = f(n) + 5 \]

So the correct answer is:
\[ f(n + 1) = f(n) + 5 \]

The other choices do not correctly represent the given arithmetic sequence:
- \( f(n + 1) = f(n + 5) \): This would incorrectly indicate that to find the next term \( f(n + 1) \), you look 5 terms ahead.
- \( f(n + 1) = 5f(n) \): This suggests a geometric sequence where each term is 5 times the previous term, which is not the case here.
- \( f(n + 1) = f(5n) \): This would suggest a very different relationship, where the next term is based on a much later term in the sequence, and again does not reflect the arithmetic increase by 5.