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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].

Sure! Let's analyze the problem step by step:

We're given a sequence: [tex]$14, 24, 34, 44, 54, \ldots$[/tex], which is an arithmetic sequence. This means that the difference between consecutive terms is constant.

Step 1: Find 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 [tex]$10$[/tex].

Step 2: Identify the first term.

The first term of the sequence is clearly given as [tex]$14$[/tex].

Step 3: Write the recursive function.

A recursive function for an arithmetic sequence with a common difference [tex]$d$[/tex] and a first term [tex]$f(1)$[/tex] is generally expressed as:
[tex]\[
f(n+1) = f(n) + d
\][/tex]

Given that:
- The common difference [tex]$d = 10$[/tex]
- The first term [tex]$f(1) = 14$[/tex]

The recursive function is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]

Conclusion:

The correct statement that describes the recursive function used to generate the 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]."