Question

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

f(1) = 6, f(4) = 12, f(7) = 18

Answer 1

To determine the recursive formula for the given arithmetic sequence, we need to find the common difference \(d\) and the relationship between consecutive terms.

Let's summarize the given terms:
- \(f(1) = 6\)
- \(f(4) = 12\)
- \(f(7) = 18\)

In an arithmetic sequence, the difference between consecutive terms is constant.

First, we find the differences between the terms we have:

1. From \(f(1)\) to \(f(4)\):
\[
d = \frac{f(4) - f(1)}{4 - 1} = \frac{12 - 6}{3} = \frac{6}{3} = 2
\]

2. From \(f(4)\) to \(f(7)\):
\[
d = \frac{f(7) - f(4)}{7 - 4} = \frac{18 - 12}{3} = \frac{6}{3} = 2
\]

Both calculations give us the same common difference, \(d = 2\).

In an arithmetic sequence, each term is the previous term plus the common difference \(d\).

The recursive formula can be written as:
\[
f(n) = f(n-1) + d
\]

Since we already know the common difference \(d = 2\), the recursive formula for the sequence is:
\[
f(n) = f(n-1) + 2
\]

Thus, the recursive formula defining the given arithmetic sequence is:
\[
f(n) = f(n-1) + 2
\]