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 identifying the recursive function used to generate the given arithmetic sequence, we need to first understand the characteristics of an arithmetic sequence. An arithmetic sequence has a constant difference between consecutive terms, known as the common difference.

Here’s how we can determine the correct recursive function statement:

1. Identify the given sequence:
The sequence provided is: 14, 24, 34, 44, 54, ...

2. Determine the common difference:
To find the common difference, subtract the first term from the second term:
[tex]\[
24 - 14 = 10
\][/tex]
This indicates that each term in the sequence increases by 10 from the previous term.

3. Identify the first term:
The first term of the sequence is 14.

4. Choose the correct recursive function:
Now, we need to find the statement that correctly describes this sequence using its first term and common difference. The recursive formula for an arithmetic sequence can be expressed 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.

Given our findings:
- The common difference is 10.
- The first term is 14.

The correct statement 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 function accurately describes the way each term in the sequence is generated by adding 10 to the previous term, starting from the first term of 14.