To find the recursive function for the sequence \(41, 29, 17, 5\), we first need to observe the differences between consecutive terms.
-
From 41 to 29:
\(29 - 41 = -12\) -
From 29 to 17:
\(17 - 29 = -12\) -
From 17 to 5:
\(5 - 17 = -12\)
The common difference is \(-12\) for each step.
This suggests that the next term can be obtained by subtracting \(12\) from the previous term. In a recursive function format, we can express this as:
\[ a_n = a_{n-1} - 12 \]
Now, looking at the provided options, however, none of them follow this exact format. Let's examine each option you provided:
A. \(a_n = a_{n-1} + 12.53\) - This does not match.
B. \(a_n = a_{n-1} - 12.41\) - This does not match exactly, although it includes subtraction.
C. \(a_n = 12a_{n-1} - 12\) - This is not correct as the sequence does not multiply by 12.
From our analysis, it seems the assumptions of the options given may not perfectly convey the desired relation. However, if we consider that the correct form should involve a subtraction of integer \(12\), we can assume there is a miscommunication in the options.
To summarize, the function based on the observations is:
- Recursive function: \(a_n = a_{n-1} - 12\)
- Initial term: \(a_1 = 41\)
Since none of the options fit the exact recursive relationship we derived, it's possible that a mistake exists in the choices you have or in the interpretation of the question. But the correct interpretation aligns with \(a_n = a_{n-1} - 12\) with \(a_1 = 41\).