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.
-
First term (\(a_1\)): The first term of the sequence is \(a_1 = 12\).
-
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 \(d = \frac{4}{3}\).
-
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: \[ \boxed{a_n = \frac{4n + 32}{3}} \]