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(2)=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 solve this problem, we need to analyze the arithmetic sequence given: 14, 24, 34, 44, 54, ...

1. Identify the Common Difference:
- In an arithmetic sequence, the difference between consecutive terms is constant. This is known as 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.

2. Create the Recursive Function:
- A recursive function for an arithmetic sequence can be expressed as:
[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]
- Here, we already determined the common difference is 10.

3. Determine the Initial Term:
- The first term of the sequence is given as 14, which is represented as [tex]$ f(1) = 14 $[/tex].

4. Select the Correct Statement:
- From the given statements, the one that correctly describes the recursive function for this 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]."

This statement accurately captures the recursive function necessary to generate the sequence 14, 24, 34, 44, 54, etc.