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------------------------------------------------ The pattern of numbers below is an arithmetic sequence:

[tex]\[ 14, 24, 34, 44, 54, \ldots \][/tex]

Which statement describes the recursive function used to generate the sequence?

A. The common difference is 1, so the function is [tex] f(n+1) = f(n) + 1 [/tex] where [tex] f(1) = 14 [/tex].

B. The common difference is 4, so the function is [tex] f(n+1) = f(n) + 4 [/tex] where [tex] f(1) = 10 [/tex].

C. The common difference is 10, so the function is [tex] f(n+1) = f(n) + 10 [/tex] where [tex] f(1) = 14 [/tex].

D. The common difference is 14, so the function is [tex] f(n+1) = f(n) + 14 [/tex] where [tex] f(1) = 10 [/tex].

To determine the recursive function used to generate the given arithmetic sequence, let's analyze the sequence step by step.

The sequence is: [tex]$14, 24, 34, 44, 54, \ldots$[/tex].

1. Identify the common difference:
- To find the common difference in an arithmetic sequence, subtract a term from the next term.
- For this sequence, subtract the first term from the second term: [tex]$24 - 14 = 10$[/tex].
- This pattern continues across the sequence, so the common difference is [tex]$10$[/tex].

2. Determine the starting point:
- The first term of the sequence is [tex]$14$[/tex]. Hence, [tex]$f(1) = 14$[/tex].

3. Write the recursive formula:
- A recursive function for an arithmetic sequence is expressed as:

[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]

- For our sequence, this becomes:

[tex]\[
f(n+1) = f(n) + 10
\][/tex]

where the initial term [tex]$f(1) = 14$[/tex].

4. Conclusion:
- The correct statement describing the recursive function is: "The common difference is 10, so the function is [tex]$f(n+1) = f(n) + 10$[/tex] where [tex]$f(1) = 14$[/tex]."

This step-by-step breakdown provides a clear understanding of how the recursive function is derived from the sequence.