- The problem asks for the recursive formula of an arithmetic sequence where \(5\) is added to the current term to get the next term.
- The term \(f(n)\) represents the current term, and \(f(n+1)\) represents the next term.
- The statement "adding 5 to each term to determine each successive term" is translated to the equation \(f(n+1) = f(n) + 5\).
- Comparing this to the given choices, the correct formula is \(\boxed{f(n+1)=f(n)+5}\).
Explanation
- Analyzing the recursive sequence definition
The problem asks to identify the recursive formula for an arithmetic sequence where each term is found by adding \(5\) to the preceding term.
In a recursive formula, \(f(n)\) typically denotes the \(n\)-th term of the sequence, and \(f(n+1)\) represents the next (or successor) term. The instruction "adding 5 to each term to determine each successive term" means the next term is equal to the current term plus 5. Mathematically, this is expressed as:
\[f(n+1) = f(n) + 5\]
- Evaluating the given options
We evaluate the given options:
- \(f(n+1) = f(n) + 5\): This formula states the next term is obtained by adding \(5\) to the current term. This matches our requirement perfectly.
- \(f(n+1) = f(n+5)\): This indicates the next term is equal to the term at the position \(n+5\), which does not describe adding \(5\) to the value of the term.
- \(f(n+1) = 5f(n)\): This formula represents a geometric sequence where the next term is \(5\) times the current term, not adding \(5\).
- \(f(n+1) = f(5n)\): This formula states the next term is equal to the term at the position \(5n\), which does not involve adding \(5\) to the value of the term.
Therefore, the only correct formula is \(f(n+1) = f(n) + 5\).
- Conclusion
Based on the analysis, the formula that represents adding \(5\) to each term to determine the next term is \(f(n+1)=f(n)+5\). The correct option is the first one. The final answer is \(\boxed{f(n+1)=f(n)+5}\).
Examples
Recursive formulas are essential in computer programming for defining functions that call themselves and in finance for calculating compound interest or scheduled payments. For example, if you save an extra \$5 every week in a piggy bank starting from a set amount, the amount in the bank at week \(n+1\) would be \(f(n+1) = f(n) + 5\). This simple model allows you to project your future savings easily without calculating every week individually.