Answer :
- The common difference of the arithmetic sequence is found to be $10$.
- The recursive function is of the form $f(n+1) = f(n) + d$, where $d$ is the common difference.
- Substituting the common difference, the recursive function is $f(n+1) = f(n) + 10$.
- The first term is $f(1) = 14$, so the recursive function is $f(n+1) = f(n) + 10$ where $f(1) = 14$.
### Explanation
1. Understanding the Problem
We are given an arithmetic sequence $14, 24, 34, 44, 54, \ldots$ and asked to find the recursive function that generates this sequence. A recursive function defines each term in the sequence based on the previous term. In an arithmetic sequence, the difference between consecutive terms is constant. This constant difference is called the common difference.
2. Finding the Common Difference
To find the common difference, we subtract any term from its subsequent term. For example, $24 - 14 = 10$, $34 - 24 = 10$, $44 - 34 = 10$, and $54 - 44 = 10$. Thus, the common difference is $10$.
3. Writing the Recursive Function
The general form of a recursive function for an arithmetic sequence is $f(n+1) = f(n) + d$, where $d$ is the common difference. In our case, $d = 10$, so the recursive function is $f(n+1) = f(n) + 10$.
4. Defining the Initial Condition
The first term of the sequence is $14$, so $f(1) = 14$. Therefore, the complete recursive function is $f(n+1) = f(n) + 10$ where $f(1) = 14$.
5. Final Answer
The correct statement describing the recursive function is: The common difference is 10, so the function is $f(n+1) = f(n) + 10$ where $f(1) = 14$.
### Examples
Recursive functions are used in computer science to define functions that call themselves. For example, calculating the factorial of a number can be done recursively. Understanding recursive functions helps in designing algorithms for various computational tasks, such as searching, sorting, and tree traversal.
- The recursive function is of the form $f(n+1) = f(n) + d$, where $d$ is the common difference.
- Substituting the common difference, the recursive function is $f(n+1) = f(n) + 10$.
- The first term is $f(1) = 14$, so the recursive function is $f(n+1) = f(n) + 10$ where $f(1) = 14$.
### Explanation
1. Understanding the Problem
We are given an arithmetic sequence $14, 24, 34, 44, 54, \ldots$ and asked to find the recursive function that generates this sequence. A recursive function defines each term in the sequence based on the previous term. In an arithmetic sequence, the difference between consecutive terms is constant. This constant difference is called the common difference.
2. Finding the Common Difference
To find the common difference, we subtract any term from its subsequent term. For example, $24 - 14 = 10$, $34 - 24 = 10$, $44 - 34 = 10$, and $54 - 44 = 10$. Thus, the common difference is $10$.
3. Writing the Recursive Function
The general form of a recursive function for an arithmetic sequence is $f(n+1) = f(n) + d$, where $d$ is the common difference. In our case, $d = 10$, so the recursive function is $f(n+1) = f(n) + 10$.
4. Defining the Initial Condition
The first term of the sequence is $14$, so $f(1) = 14$. Therefore, the complete recursive function is $f(n+1) = f(n) + 10$ where $f(1) = 14$.
5. Final Answer
The correct statement describing the recursive function is: The common difference is 10, so the function is $f(n+1) = f(n) + 10$ where $f(1) = 14$.
### Examples
Recursive functions are used in computer science to define functions that call themselves. For example, calculating the factorial of a number can be done recursively. Understanding recursive functions helps in designing algorithms for various computational tasks, such as searching, sorting, and tree traversal.
