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 solve the problem of identifying the recursive function for the given arithmetic sequence [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex], we need to follow these steps:

1. Identify the Common Difference:
The sequence is arithmetic, which means the difference between consecutive terms is constant. To find this common difference, subtract any term from its subsequent term.
[tex]\[
24 - 14 = 10
\][/tex]
Therefore, the common difference is 10.

2. Identify the First Term:
The first term of the sequence is given as 14.

3. Formulate the Recursive Function:
A recursive function for an arithmetic sequence can be described as:
[tex]\[
f(n+1) = f(n) + d
\][/tex]
where [tex]\(d\)[/tex] is the common difference.

Using the values we identified:
- The common difference [tex]\(d\)[/tex] is 10.
- The first term [tex]\(f(1)\)[/tex] is 14.

We can then express the recursive function as:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]

This matches with the correct statement:
"The common difference is 10, so the function is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 14\)[/tex]."