Three terms of an arithmetic sequence are shown below. Which recursive formula defines the sequence?

To determine which recursive formula defines the arithmetic sequence, let's first identify the common difference of the sequence.

We are given:

- \( f(1) = 6 \)
- \( f(4) = 12 \)
- \( f(7) = 18 \)

First, calculate the number of terms between consecutive given terms:

- From \( f(1) \) to \( f(4) \), there are 3 terms between them.
- From \( f(4) \) to \( f(7) \), there are also 3 terms between them.

Given that an arithmetic sequence advances by adding a common difference each time, and each interval consists of 3 steps, we can infer that the increase from one given term to the next is divided among 3 steps.

Let's compute the common difference:

1. The difference between \( f(4) \) and \( f(1) \):
\( 12 - 6 = 6 \)
Because there are 3 steps between \( f(1) \) and \( f(4) \), the common difference \( d \) is:
\( d = \frac{6}{3} = 2 \)

2. Likewise, the difference between \( f(7) \) and \( f(4) \):
\( 18 - 12 = 6 \)
Again, because there are 3 steps between \( f(4) \) and \( f(7) \), the common difference \( d \) remains:
\( d = \frac{6}{3} = 2 \)

Hence, each term in the sequence increases by 2 from the previous term.

Therefore, the recursive formula for the sequence, given the common difference \( d = 2 \), is:

\[ f(n + 1) = f(n) + 2 \]

Thus, the correct recursive formula is:

\[ f (n + 1) = f(n) + 2 \]