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 this sequence?

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

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

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

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

To determine the recursive function used in the given arithmetic sequence, follow these steps:

1. Identify the Sequence:
The sequence provided is: 14, 24, 34, 44, 54, ...

2. Find the Common Difference:
An arithmetic sequence has a constant difference (known as the common difference) between consecutive terms. Let’s calculate it:
- The first term is 14.
- The second term is 24.

To find the common difference, subtract the first term from the second term:
[tex]\[
\text{Common Difference} = 24 - 14 = 10
\][/tex]

3. Formulate the Recursive Function:
In an arithmetic sequence, the recursive formula is expressed as:
[tex]\[
f(n+1) = f(n) + \text{Common Difference}
\][/tex]

Substituting the common difference we found:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]

4. Choose the Correct Statement:
From the options given:
- The common difference is 1, so the function is [tex]$f(n+1) = f(n) + 1$[/tex]
- The common difference is 4, so the function is [tex]$f(n+1) = f(n) + 4$[/tex]
- The common difference is 10, so the function is [tex]$f(n+1) = f(n) + 10$[/tex]
- The common difference is 14, so the function is [tex]$f(n+1) = f(n) + 14$[/tex]

The correct statement is that the common difference is 10, thus the function is:
[tex]\[
f(n+1) = f(n) + 10
\][/tex]

So, the correct recursive function for this arithmetic sequence is [tex]$f(n+1) = f(n) + 10$[/tex].