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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 solve this question, we need to determine the correct statement about the recursive function used to generate the given arithmetic sequence: 14, 24, 34, 44, 54, ...

1. Identify the Common Difference: An arithmetic sequence is defined by a constant difference between consecutive terms. To find this common difference, we subtract the first term from the second term:

[tex]\[
24 - 14 = 10
\][/tex]

So, the common difference is 10.

2. Write the Recursive Function: In a recursive function for an arithmetic sequence, each term is generated by adding the common difference to the previous term. The recursive function can be written as:

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

Here, [tex]\( f(n) \)[/tex] is the current term of the sequence, and [tex]\( f(n+1) \)[/tex] is the next term.

3. Determine the Initial Value: The initial value for the recursive function is the first term of the sequence, which is 14. Therefore, we set:

[tex]\[
f(1) = 14
\][/tex]

Putting it all together, the correct statement about 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]."