Answer :
To solve this problem, we need to determine the recursive function that describes the arithmetic sequence provided: 14, 24, 34, 44, 54, ...
1. Identify the Pattern:
The sequence is an arithmetic sequence, which means it has a constant difference between each term.
2. Find the Common Difference:
To find the common difference, subtract the first term from the second term:
[tex]\[
24 - 14 = 10
\][/tex]
So, the common difference is 10.
3. Determine the Recursive Function:
In an arithmetic sequence, if the common difference is [tex]\(d\)[/tex], and the first term is [tex]\(f(1)\)[/tex], the recursive formula is given by:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
In this sequence, the common difference [tex]\(d\)[/tex] is 10, and the first term [tex]\(f(1)\)[/tex] is 14. Therefore, the recursive function is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
4. Evaluate the Given Statements:
- Option 1: 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.
- Option 2: 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 neither the common difference nor the first term is correct.
- Option 3: 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 the identified common difference and first term.
- Option 4: The common difference is 14, so the function is [tex]\(f(n+1) = f(n) + 14\)[/tex] where [tex]\(f(1) = 10\)[/tex].
- This is incorrect because neither the common difference nor the first term is correct.
Therefore, the correct statement that describes the recursive function used to generate the sequence is: "The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex]."
1. Identify the Pattern:
The sequence is an arithmetic sequence, which means it has a constant difference between each term.
2. Find the Common Difference:
To find the common difference, subtract the first term from the second term:
[tex]\[
24 - 14 = 10
\][/tex]
So, the common difference is 10.
3. Determine the Recursive Function:
In an arithmetic sequence, if the common difference is [tex]\(d\)[/tex], and the first term is [tex]\(f(1)\)[/tex], the recursive formula is given by:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
In this sequence, the common difference [tex]\(d\)[/tex] is 10, and the first term [tex]\(f(1)\)[/tex] is 14. Therefore, the recursive function is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]
4. Evaluate the Given Statements:
- Option 1: 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.
- Option 2: 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 neither the common difference nor the first term is correct.
- Option 3: 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 the identified common difference and first term.
- Option 4: The common difference is 14, so the function is [tex]\(f(n+1) = f(n) + 14\)[/tex] where [tex]\(f(1) = 10\)[/tex].
- This is incorrect because neither the common difference nor the first term is correct.
Therefore, the correct statement that describes the recursive function used to generate the sequence is: "The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex]."
