To solve this problem, let's break it down step-by-step:
1. Understanding an Arithmetic Sequence:
In an arithmetic sequence, each term after the first is formed by adding a constant difference to the previous term. For this case, we are given that 5 is added to each term to get the next term.
2. Identifying the Correct Recursive Formula:
- A recursive formula for an arithmetic sequence usually expresses the next term [tex]\( f(n+1) \)[/tex] in terms of the previous term [tex]\( f(n) \)[/tex] plus a constant difference.
- Since 5 is the constant difference given in the problem, we want a formula where [tex]\( f(n+1) \)[/tex] is the previous term [tex]\( f(n) \)[/tex] plus 5.
3. Evaluating the Given Options:
Let's evaluate each provided option to identify which one correctly represents this pattern:
- [tex]\( f(n+1) = f(n) + 5 \)[/tex]:
This formula reads as "the next term is the previous term plus 5," which matches our understanding of the given arithmetic sequence.
- [tex]\( f(n+1) = f(n + 5) \)[/tex]:
This would imply that the next term is equal to the term five positions ahead, which is not consistent with adding a constant difference of 5 to get the next term.
- [tex]\( f(n+1) = 5 f(n) \)[/tex]:
This would imply that the next term is five times the previous term, which is a geometric progression, not an arithmetic one.
- [tex]\( f(n+1) = f(5n) \)[/tex]:
This suggests the next term is some term that is dependent linearly on the position [tex]\( n \)[/tex], multiplied by 5. This does not match the given arithmetic sequence pattern either.
4. Conclusion:
The correct recursive formula that represents the arithmetic sequence where 5 is added to each term to get the next term is:
[tex]\[
f(n+1) = f(n) + 5
\][/tex]
This matches our expected pattern for an arithmetic sequence with a common difference of 5.
