College

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].

Answer :

To determine the recursive function used to generate the given arithmetic sequence, we need to identify the common difference and the first term of the sequence.

Here's the sequence: [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex]

### Step-by-Step Solution:

1. Identify the First Term:
- The first term of the sequence, denoted as [tex]\(f(1)\)[/tex], is the first number in the sequence.
- Here, [tex]\(f(1) = 14\)[/tex].

2. Find the Common Difference:
- An arithmetic sequence has a constant difference between consecutive terms. 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].

3. Write the Recursive Function:
- A recursive function for an arithmetic sequence can be written as [tex]\(f(n+1) = f(n) + d\)[/tex], where [tex]\(d\)[/tex] is the common difference and [tex]\(f(1)\)[/tex] is the first term.
- Based on our findings, the common difference is [tex]\(10\)[/tex], and the first term is [tex]\(14\)[/tex].
- Therefore, the recursive function is:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]

Thus, the correct statement that describes the recursive function for 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]."