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 recursive formula for arithmetic sequence

In an arithmetic sequence, the recursive formula is of the form \( f(n + 1)=f(n)+d \), where \( d \) is the common difference (the amount added to each term to get the next term).

Step2: Identify the common difference

Here, the problem states that 5 is added to each term to get the successive term, so \( d = 5 \).

Step3: Match with the given options

• For the first option \( f(n + 1)=f(n)+5 \), it follows the form of the arithmetic sequence recursive formula with \( d = 5 \).
• The second option \( f(n + 1)=f(n + 5) \) is not the correct form as it is not adding 5 to the previous term but rather using \( n+5 \) as the input, which is not the definition of an arithmetic sequence.
• The third option \( f(n + 1)=5f(n) \) represents a geometric sequence (multiplying by 5), not an arithmetic sequence (which requires addition).
• The fourth option \( f(n + 1)=f(5n) \) is also not the correct form for an arithmetic sequence as it is using \( 5n \) as the input, not related to adding 5 to the previous term.

Answer:

\( f(n + 1)=f(n)+5 \) (the first option among the given choices)