- The function notation \(f(n) = x\) denotes that at position \(n\), the value of the function is \(x\).
- Applying this to sequences, \(f(3)\) refers to the 3rd term, and the equality \(f(3) = 1\) means the value of this 3rd term is 1.
- Comparing this to the given options, the correct statement is that the third term in the sequence has a value of 1.
- The final answer is \(\boxed{d}\).
Explanation
- Understanding Function Notation in Sequences
The notation \(f(n) = x\) is standard in mathematics to describe a function \(f\), where \(n\) represents the input (also known as the argument) and \(x\) represents the corresponding output (the value of the function). When we apply this notation to a sequence defined by a function, the input \(n\) refers to the position or the index of the term in the sequence, and the output \(f(n)\) refers to the numerical value of the sequence at that position.
- Analyzing the Specific Notation \(f(3) = 1\)
Given the expression \(f(3) = 1\), we identify the input \(n=3\) and the output value \(1\). In the context of sequences, an input of \(3\) indicates that we are looking at the 3rd term in the sequence. The value \(1\) indicates that the value of the term at that 3rd position is \(1\). Therefore, the notation \(f(3) = 1\) explicitly translates to "the third term in the sequence is \(1\)".
- Evaluating the Options
Now we evaluate the provided options against our analysis:
- Option (a): "The common difference of the sequence is 3." This would imply an arithmetic progression property, not the specific meaning of \(f(3)\).
- Option (b): "The first term in the sequence has a value of 3." This would correspond to \(f(1) = 3\).
- Option (c): "The common ratio of the sequence is 3." This would imply a geometric progression property, not the specific meaning of \(f(3)\).
- Option (d): "The third term in the sequence has a value of 1." This aligns perfectly with our finding that the 3rd term is \(1\).
Conclusion: The correct option is (d).
Examples
In a classroom setting, a teacher might present a sequence of seat numbers and their corresponding student IDs. If the function is defined as \(f(n)\), where \(n\) is the seat number, then \(f(3) = 1\) would simply mean that the student sitting in seat number 3 has the student ID number 1. This helps students relate abstract mathematical notation to real-world data tracking and labeling systems.