To find a recursive formula for the sequence \(a_n\) represented by the terms \(8, 2, -4, -10, \ldots\), we first need to observe the pattern in the sequence.
- The first term is \(a_1 = 8\).
- The second term is \(a_2 = 2\).
- The third term is \(a_3 = -4\).
- The fourth term is \(a_4 = -10\).
Next, let's calculate the differences between consecutive terms:
- \(a_2 - a_1 = 2 - 8 = -6\)
- \(a_3 - a_2 = -4 - 2 = -6\)
- \(a_4 - a_3 = -10 - (-4) = -6\)
We can see that each term is obtained by subtracting \(6\) from the previous term. Therefore, we can express the recursive relationship as follows:
\[ a_n = a_{n-1} - 6 \quad \text{for } n \geq 2 \]
We also need to provide the initial condition:
\[ a_1 = 8 \]
Thus, the complete recursive formula for the sequence is:
\[ \begin{align*} a_1 & = 8 \ a_n & = a_{n-1} - 6 \quad \text{for } n \geq 2 \end{align*} \]