To determine the correct recursive function used to generate the sequence [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex], we need to find the common difference and initial value, and then use these to check each provided statement.
1. Calculate the common difference:
- Take any two consecutive terms in the sequence.
- Subtract the first term from the second term: [tex]\(24 - 14 = 10\)[/tex].
- Confirm by checking other pairs: [tex]\(34 - 24 = 10\)[/tex], [tex]\(44 - 34 = 10\)[/tex], [tex]\(54 - 44 = 10\)[/tex].
- Therefore, the common difference is 10.
2. Identify the initial value:
- The first term of the sequence is [tex]\(14\)[/tex].
3. Check each statement against the identified common difference and initial value:
- Statement 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 statement is incorrect because the common difference is not 1.
- Statement 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 statement is incorrect because both the common difference and initial value are wrong.
- Statement 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 statement is correct because the common difference is 10 and the initial value is 14.
- Statement 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 statement is incorrect because both the common difference and initial value are wrong.
Therefore, the correct statement that describes the recursive function used to generate the sequence [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex] is:
The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex].
