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) )

Explanation:

Step1: Recall Arithmetic Sequence Recursion

An arithmetic sequence has a common difference \( d \), and the recursive formula is \( f(n + 1)=f(n)+d \), where \( d \) is the common difference.

Step2: Identify the Common Difference

Here, the common difference \( d = 5 \) (since 5 is added to each term to get the next term). So the recursive formula should be \( f(n + 1)=f(n)+5 \).

Step3: Analyze Other Options

• \( f(n + 1)=f(n + 5) \): This does not represent adding 5 to the previous term, but rather taking the term 5 positions ahead, which is incorrect.
• \( f(n + 1)=5f(n) \): This is a geometric sequence formula (multiplying by 5), not arithmetic.
• \( f(n + 1)=f(5n) \): This is not a standard arithmetic or geometric recursion, and does not represent adding 5 to the previous term.

Answer:

A. \( f(n + 1) = f(n) + 5 \)