Question

Find an explicit formula for the arithmetic sequence below.
-12,-35/3,34/3

Answer 1

To find an explicit formula for the arithmetic sequence given, we first need to determine the first term and the common difference.

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

2. **Calculate the common difference (\(d\))**:
- The second term is \(a_2 = -\frac{35}{3}\).
- The common difference is calculated as follows:
\[
d = a_2 - a_1 = -\frac{35}{3} - \left(-12\right)
\]
- To subtract \(-12\), we convert it to a fraction with a denominator of 3:
\[
-12 = -\frac{36}{3}
\]
- Now we can calculate \(d\):
\[
d = -\frac{35}{3} + \frac{36}{3} = \frac{1}{3}
\]

Now we have \(a_1 = -12\) and \(d = \frac{1}{3}\).

3. **Write the explicit formula**:
The explicit formula for an arithmetic sequence is given by:
\[
a_n = a_1 + (n-1)d
\]
Substituting the values of \(a_1\) and \(d\):
\[
a_n = -12 + (n-1) \cdot \frac{1}{3}
\]

4. **Simplify the formula**:
\[
a_n = -12 + \frac{n-1}{3} = -12 + \frac{n}{3} - \frac{1}{3} = -\frac{36}{3} - \frac{1}{3} + \frac{n}{3} = \frac{n - 37}{3}
\]

Thus, the explicit formula for the arithmetic sequence is:
\[
a_n = \frac{n - 37}{3}
\]