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 solve the problem of determining the recursive function for the given arithmetic sequence, let's break it down step-by-step:

1. Identify the Sequence:
The sequence given is:
[tex]\[14, 24, 34, 44, 54, \ldots\][/tex]

2. Determine the Common Difference:
Since this is an arithmetic sequence, the difference between consecutive terms is constant.
To find this difference, subtract the first term from the second term:
[tex]\[24 - 14 = 10\][/tex]

3. Write the Recursive Function:
An arithmetic sequence can be described by the recursive function:
[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 of the sequence.

4. Apply the Identified Values:
- The common difference [tex]\(d\)[/tex] is 10.
- The first term [tex]\(f(1)\)[/tex] is 14.

Replacing these values into the recursive function gives us:
[tex]\[f(n+1) = f(n) + 10\][/tex]
with [tex]\(f(1) = 14\)[/tex].

Therefore, 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]."