Question

Jessie wrote the arithmetic sequence of numbers -18, -13, -8, -3, 2, 7.

Which rule can be used to model this sequence of numbers?

Answer 1

An arithmetic sequence is defined by a common difference between consecutive terms. In this sequence, we can find the common difference:

- The difference between the second term (-13) and the first term (-18) is:
\(-13 - (-18) = -13 + 18 = 5\)

- The difference between the third term (-8) and the second term (-13) is:
\(-8 - (-13) = -8 + 13 = 5\)

- The difference between the fourth term (-3) and the third term (-8) is:
\(-3 - (-8) = -3 + 8 = 5\)

- The difference between the fifth term (2) and the fourth term (-3) is:
\(2 - (-3) = 2 + 3 = 5\)

- The difference between the sixth term (7) and the fifth term (2) is:
\(7 - 2 = 5\)

Since the common difference is consistently 5, we can conclude that the rule for the arithmetic sequence can be expressed as:

\[ a_n = a_1 + (n - 1)d \]

Where:
- \( a_n \) is the nth term of the sequence,
- \( a_1 \) is the first term,
- \( n \) is the term number,
- \( d \) is the common difference.

In this case:
- The first term \( a_1 = -18 \)
- The common difference \( d = 5 \)

Thus, the rule for this sequence is:

\[
a_n = -18 + (n - 1) \cdot 5
\]

Alternatively, this can be simplified as:

\[
a_n = -18 + 5n - 5
\]
\[
a_n = 5n - 23
\]

So, the rule that models the sequence is:

\[
a_n = 5n - 23
\]