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Answer :
- The given sequence is an arithmetic sequence with a first term of 14.
- The common difference is calculated by subtracting consecutive terms: $24 - 14 = 10$.
- The recursive function is defined as $f(n+1) = f(n) + d$, where $d$ is the common difference.
- Therefore, the recursive function is $f(n+1) = f(n) + 10$ with $f(1) = 14$.
### Explanation
1. Analyzing the Sequence
Let's analyze the given arithmetic sequence: $14, 24, 34, 44, 54, \ldots$. We need to find the recursive function that generates this sequence. A recursive function defines each term in the sequence based on the previous term.
2. General 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. The first term is $f(1) = 14$.
3. Finding the Common Difference
To find the common difference $d$, we subtract consecutive terms. For example, $24 - 14 = 10$. So, the common difference is $d = 10$.
4. Writing the Recursive Function
Now we can write the recursive function as $f(n+1) = f(n) + 10$, where $f(1) = 14$. This means that to find the next term in the sequence, we add 10 to the previous term, starting with the first term 14.
5. Conclusion
Therefore, the correct statement is: The common difference is 10, so the function is $f(n+1) = f(n) + 10$ where $f(1) = 14$.
### Examples
Arithmetic sequences and recursive functions are used in various real-life scenarios, such as calculating simple interest on a savings account, predicting population growth (under simplified conditions), or determining the number of seats in rows of a stadium where each row has a fixed number of additional seats compared to the previous row. Understanding these concepts helps in making predictions and managing resources effectively.
- The common difference is calculated by subtracting consecutive terms: $24 - 14 = 10$.
- The recursive function is defined as $f(n+1) = f(n) + d$, where $d$ is the common difference.
- Therefore, the recursive function is $f(n+1) = f(n) + 10$ with $f(1) = 14$.
### Explanation
1. Analyzing the Sequence
Let's analyze the given arithmetic sequence: $14, 24, 34, 44, 54, \ldots$. We need to find the recursive function that generates this sequence. A recursive function defines each term in the sequence based on the previous term.
2. General 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. The first term is $f(1) = 14$.
3. Finding the Common Difference
To find the common difference $d$, we subtract consecutive terms. For example, $24 - 14 = 10$. So, the common difference is $d = 10$.
4. Writing the Recursive Function
Now we can write the recursive function as $f(n+1) = f(n) + 10$, where $f(1) = 14$. This means that to find the next term in the sequence, we add 10 to the previous term, starting with the first term 14.
5. Conclusion
Therefore, the correct statement is: The common difference is 10, so the function is $f(n+1) = f(n) + 10$ where $f(1) = 14$.
### Examples
Arithmetic sequences and recursive functions are used in various real-life scenarios, such as calculating simple interest on a savings account, predicting population growth (under simplified conditions), or determining the number of seats in rows of a stadium where each row has a fixed number of additional seats compared to the previous row. Understanding these concepts helps in making predictions and managing resources effectively.
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