High School

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 solve the question about the pattern of numbers in the arithmetic sequence [tex]\(14, 24, 34, 44, 54, \ldots\)[/tex], let's break it down step-by-step:

1. Understand the Arithmetic Sequence:
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the "common difference."

2. Identify the Common Difference:
To find the common difference, subtract any term from the term that follows it. Let's use the first two terms in the sequence:
[tex]\[
24 - 14 = 10
\][/tex]
So, the common difference is [tex]\(10\)[/tex].

3. Writing the Recursive Formula:
A recursive formula defines each term of the sequence by referring to the previous term. For an arithmetic sequence, the recursive formula is generally expressed as:
[tex]\[
f(n+1) = f(n) + \text{common difference}
\][/tex]
For the given sequence, since we determined that the common difference is [tex]\(10\)[/tex] and the first term [tex]\(f(1)\)[/tex] is [tex]\(14\)[/tex], the recursive function can be written as:
[tex]\[
f(n+1) = f(n) + 10 \quad \text{where} \quad f(1) = 14
\][/tex]

4. Putting it All Together:
Based on the steps and the arithmetic sequence given, the correct statement describing 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].

This recursive formula correctly describes how to generate each term of the sequence from the previous one.