Question

Find an explicit formula for the arithmetic sequence below.
12,40/3,44/3,16

Answer 1

To find an explicit formula for the given arithmetic sequence \(12, \frac{40}{3}, \frac{44}{3}, 16\), we first 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 (\(d\))**: We can find the common difference by subtracting the first term from the second term:
\[
d = a_2 - a_1 = \frac{40}{3} - 12
\]
To perform this subtraction, we need to express 12 in terms of thirds:
\[
12 = \frac{36}{3}
\]
Now we can compute the difference:
\[
d = \frac{40}{3} - \frac{36}{3} = \frac{4}{3}
\]
We will check the common difference between other adjacent terms to confirm:
\[
a_3 - a_2 = \frac{44}{3} - \frac{40}{3} = \frac{4}{3}
\]
\[
a_4 - a_3 = 16 - \frac{44}{3}
\]
Again, express 16 in thirds:
\[
16 = \frac{48}{3}
\]
Now calculate:
\[
a_4 - a_3 = \frac{48}{3} - \frac{44}{3} = \frac{4}{3}
\]
Since the common difference \(d\) is consistent, we have verified \(d = \frac{4}{3}\).

3. **Explicit formula**: The explicit formula for an arithmetic sequence is given by:
\[
a_n = a_1 + (n - 1) d
\]
Substituting the values we found:
\[
a_n = 12 + (n - 1) \cdot \frac{4}{3}
\]
Simplifying this:
\[
a_n = 12 + \frac{4(n - 1)}{3} = 12 + \frac{4n - 4}{3}
\]
To combine terms, express 12 as a fraction:
\[
12 = \frac{36}{3} \implies a_n = \frac{36}{3} + \frac{4n - 4}{3} = \frac{36 + 4n - 4}{3} = \frac{4n + 32}{3}
\]

Thus, the explicit formula for the arithmetic sequence is:
\[
\boxed{a_n = \frac{4n + 32}{3}}
\]