To determine the correct recursive formula for the given arithmetic sequence, we need to identify the common difference [tex]\( d \)[/tex]. This difference is the constant amount added to each term to get the next term in an arithmetic sequence.
The sequence given is:
[tex]\[ f(1) = 6, \][/tex]
[tex]\[ f(4) = 12, \][/tex]
[tex]\[ f(7) = 18. \][/tex]
First, calculate the common difference [tex]\( d \)[/tex] using the terms provided.
1. Calculate the common difference between [tex]\( f(4) \)[/tex] and [tex]\( f(1) \)[/tex]:
[tex]\[
d = \frac{f(4) - f(1)}{4 - 1} = \frac{12 - 6}{4 - 1} = \frac{6}{3} = 2.
\][/tex]
2. Verify this common difference by checking it with another pair of terms, such as [tex]\( f(7) \)[/tex] and [tex]\( f(4) \)[/tex]:
[tex]\[
d_{\text{check}} = \frac{f(7) - f(4)}{7 - 4} = \frac{18 - 12}{7 - 4} = \frac{6}{3} = 2.
\][/tex]
Both calculations confirm that the common difference is [tex]\( d = 2 \)[/tex].
To formulate the recursive formula, recognize that in an arithmetic sequence, each term [tex]\( f(n) \)[/tex] is derived by adding the common difference [tex]\( d \)[/tex] to the previous term [tex]\( f(n-1) \)[/tex]. Therefore, the recursive formula reflecting this pattern is:
[tex]\[ f(n+1) = f(n) + 2. \][/tex]
Thus, the correct recursive formula defining this arithmetic sequence is:
[tex]\[ \boxed{f(n+1) = f(n) + 2}. \][/tex]
