Question

What kind of sequence is this?

–66, –84, –102, –120, ...

Answer 1

To determine what kind of sequence this is, we can look at the differences between the consecutive terms:

- The difference between the second term (-84) and the first term (-66) is:
\(-84 - (-66) = -84 + 66 = -18\)

- The difference between the third term (-102) and the second term (-84) is:
\(-102 - (-84) = -102 + 84 = -18\)

- The difference between the fourth term (-120) and the third term (-102) is:
\(-120 - (-102) = -120 + 102 = -18\)

As the differences between consecutive terms are constant at -18, this indicates that the sequence is an **arithmetic sequence** with a common difference of -18.

The explicit formula for the \(n\)-th term of an arithmetic sequence can be expressed as:
\[
a_n = a_1 + (n - 1)d
\]
where \(a_1\) is the first term and \(d\) is the common difference.

In this case:
- \(a_1 = -66\)
- \(d = -18\)

Thus, the \(n\)-th term of this sequence can be written as:
\[
a_n = -66 + (n - 1)(-18)
\]
This confirms that it is indeed an arithmetic sequence.