High School

[tex]$4, 24, 34, 44, 54, \ldots$[/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 correct recursive function used to generate the sequence [tex]\(4, 24, 34, 44, 54, \ldots\)[/tex], we need to find the pattern in the differences between consecutive numbers:

1. Calculate the difference between the first few terms:
- [tex]\(24 - 4 = 20\)[/tex]
- [tex]\(34 - 24 = 10\)[/tex]
- [tex]\(44 - 34 = 10\)[/tex]
- [tex]\(54 - 44 = 10\)[/tex]

2. Observe that after the first term, the common difference between consecutive numbers is consistently [tex]\(10\)[/tex].

3. This indicates the sequence follows an arithmetic pattern with a common difference of [tex]\(10\)[/tex], and the starting term [tex]\(f(1) = 4\)[/tex].

Therefore, the recursive function that describes this arithmetic sequence is [tex]\(f(n+1) = f(n) + 10\)[/tex] where [tex]\(f(1) = 4\)[/tex].

This explanation matches the sequence generated. The common difference is [tex]\(10\)[/tex], and the statement that accurately describes the sequence is:
"The common difference is 10, so the function is [tex]\(f(n+1)=f(n)+10\)[/tex] where [tex]\(f(1)=4\)[/tex]."