Answer :
Certainly! Let's examine each given formula carefully to determine which one represents the arithmetic sequence where 5 is added to each term to find the next term.
1. Option 1: \( f(n+1) = f(n) + 5 \)
- In this formula, the next term \( f(n+1) \) is equal to the current term \( f(n) \) plus 5.
- This perfectly matches the description of an arithmetic sequence where 5 is added to each term to get the next term.
- Therefore, this formula is a valid recursive representation of the sequence.
2. Option 2: \( f(n+1) = f(n+5) \)
- In this formula, the next term \( f(n+1) \) is equal to the term 5 places ahead of the current term (i.e., \( f(n+5) \)).
- This does not represent an arithmetic sequence where a constant is added to each term to find the next term.
3. Option 3: \( f(n+1) = 5 f(n) \)
- In this formula, the next term \( f(n+1) \) is 5 times the current term \( f(n) \).
- This describes a geometric sequence where the ratio between terms is 5, not an arithmetic sequence.
4. Option 4: \( f(n+1) = f(5n) \)
- In this formula, the next term \( f(n+1) \) is equal to the term at position \( 5n \).
- This does not follow the pattern of adding a constant to each term.
Given these evaluations, the correct formula that represents Shaunta's arithmetic sequence, where 5 is added to each term to determine each successive term, is:
[tex]\[ f(n+1) = f(n) + 5 \][/tex]
Therefore, the right choice is the first formula.
1. Option 1: \( f(n+1) = f(n) + 5 \)
- In this formula, the next term \( f(n+1) \) is equal to the current term \( f(n) \) plus 5.
- This perfectly matches the description of an arithmetic sequence where 5 is added to each term to get the next term.
- Therefore, this formula is a valid recursive representation of the sequence.
2. Option 2: \( f(n+1) = f(n+5) \)
- In this formula, the next term \( f(n+1) \) is equal to the term 5 places ahead of the current term (i.e., \( f(n+5) \)).
- This does not represent an arithmetic sequence where a constant is added to each term to find the next term.
3. Option 3: \( f(n+1) = 5 f(n) \)
- In this formula, the next term \( f(n+1) \) is 5 times the current term \( f(n) \).
- This describes a geometric sequence where the ratio between terms is 5, not an arithmetic sequence.
4. Option 4: \( f(n+1) = f(5n) \)
- In this formula, the next term \( f(n+1) \) is equal to the term at position \( 5n \).
- This does not follow the pattern of adding a constant to each term.
Given these evaluations, the correct formula that represents Shaunta's arithmetic sequence, where 5 is added to each term to determine each successive term, is:
[tex]\[ f(n+1) = f(n) + 5 \][/tex]
Therefore, the right choice is the first formula.
