Answer :
To solve the problem, we need to determine which statement correctly describes the recursive function for the given arithmetic sequence: 14, 24, 34, 44, 54, ...
Here's how we can work through the problem:
1. Identify the Common Difference:
- An arithmetic sequence is defined by a constant difference between consecutive terms. This constant difference is called the "common difference."
- Let's check the difference between the first two terms: [tex]\(24 - 14 = 10\)[/tex].
- To verify, check the difference between the next consecutive terms:
- [tex]\(34 - 24 = 10\)[/tex]
- [tex]\(44 - 34 = 10\)[/tex]
- [tex]\(54 - 44 = 10\)[/tex]
- All these differences confirm that the common difference is 10.
2. Determine the Recursive Function:
- A recursive function expresses each term in the sequence based on the previous term.
- Since the common difference is 10, each term is obtained by adding 10 to the previous term.
- The first term in the sequence is 14.
3. Construct the Recursive Function:
- Using the starting point and the common difference, the recursive function for this sequence is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
4. Choose the Correct Statement:
- Now, compare this recursive function with the given options:
- The common difference is 1, so the function is [tex]\(f(n+1) = f(n) + 1\)[/tex] where [tex]\(f(1) = 14\)[/tex]. (This is incorrect because the common difference is not 1).
- The common difference is 4, so the function is [tex]\(f(n+1) = f(n) + 4\)[/tex] where [tex]\(f(1) = 10\)[/tex]. (This is incorrect because the common difference is not 4).
- The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex]. (This is correct because it matches our findings).
Thus, the correct statement is: The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex].
Here's how we can work through the problem:
1. Identify the Common Difference:
- An arithmetic sequence is defined by a constant difference between consecutive terms. This constant difference is called the "common difference."
- Let's check the difference between the first two terms: [tex]\(24 - 14 = 10\)[/tex].
- To verify, check the difference between the next consecutive terms:
- [tex]\(34 - 24 = 10\)[/tex]
- [tex]\(44 - 34 = 10\)[/tex]
- [tex]\(54 - 44 = 10\)[/tex]
- All these differences confirm that the common difference is 10.
2. Determine the Recursive Function:
- A recursive function expresses each term in the sequence based on the previous term.
- Since the common difference is 10, each term is obtained by adding 10 to the previous term.
- The first term in the sequence is 14.
3. Construct the Recursive Function:
- Using the starting point and the common difference, the recursive function for this sequence is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
4. Choose the Correct Statement:
- Now, compare this recursive function with the given options:
- The common difference is 1, so the function is [tex]\(f(n+1) = f(n) + 1\)[/tex] where [tex]\(f(1) = 14\)[/tex]. (This is incorrect because the common difference is not 1).
- The common difference is 4, so the function is [tex]\(f(n+1) = f(n) + 4\)[/tex] where [tex]\(f(1) = 10\)[/tex]. (This is incorrect because the common difference is not 4).
- The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex]. (This is correct because it matches our findings).
Thus, the correct statement is: The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex].
