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:
\[ f (n + 1) = f(n) + 2 \]
Three terms of an arithmetic sequence are shown below. Which recursive formula defines the sequence?
f(1) = 6, f(4) = 12, f(7) = 18
f (n + 1) = f(n) + 6
f (n + 1) = 2f(n)
f (n + 1) = f(n) + 2
f (n + 1) = 1.5f(n)
1 answer