Question

Find an explicit formula for the arithmetic sequence below.
8,1,-6,-13

Answer 1

To find an explicit formula for the given arithmetic sequence \(8, 1, -6, -13\), we need to determine the first term and the common difference.

1. **Identify the first term (\(a_1\))**: The first term \(a_1\) is \(8\).

2. **Calculate the common difference (\(d\))**:
The common difference is found by subtracting the first term from the second term:
\[
d = a_2 - a_1 = 1 - 8 = -7
\]

We can check the common difference using the next pairs of terms:
\[
d = a_3 - a_2 = -6 - 1 = -7
\]
\[
d = a_4 - a_3 = -13 - (-6) = -13 + 6 = -7
\]
The common difference \(d = -7\) is consistent throughout the sequence.

3. **Write the explicit formula**:
The general explicit formula for the \(n\)-th term of an arithmetic sequence can be written as:
\[
a_n = a_1 + (n-1)d
\]

Substituting the values we found:
\[
a_n = 8 + (n-1)(-7)
\]

4. **Simplify the formula**:
\[
a_n = 8 - 7(n - 1)
\]
\[
a_n = 8 - 7n + 7
\]
\[
a_n = 15 - 7n
\]

Thus, the explicit formula for the arithmetic sequence is:
\[
\boxed{a_n = 15 - 7n}
\]