Question

Find an explicit formula for the arithmetic sequence below.
-12,-20,-28,-36

Answer 1

To find the explicit formula for the given arithmetic sequence \(-12, -20, -28, -36\), we first need to identify the first term and the common difference.

1. **First term (\(a_1\))**: The first term of the sequence is \(a_1 = -12\).

2. **Common difference (\(d\))**: To find the common difference, we can subtract the first term from the second term:
\[
d = a_2 - a_1 = -20 - (-12) = -20 + 12 = -8
\]
We can verify this by checking the difference between other consecutive terms:
\[
-28 - (-20) = -28 + 20 = -8 \\
-36 - (-28) = -36 + 28 = -8
\]
The common difference \(d\) is consistently \(-8\).

3. **Explicit formula**: The general formula for the \(n\)-th term of an arithmetic sequence can be expressed as:
\[
a_n = a_1 + (n - 1) d
\]
Substituting the known values \(a_1 = -12\) and \(d = -8\) into the formula:
\[
a_n = -12 + (n - 1)(-8)
\]
Simplifying the formula:
\[
a_n = -12 - 8(n - 1) \\
= -12 - 8n + 8 \\
= -8n - 4
\]

Thus, the explicit formula for the arithmetic sequence is:
\[
a_n = -8n - 4
\]