To find the explicit and recursive equations for the given arithmetic sequence, we'll start with the information provided:
- The first term \( a_1 = 2 \).
- The common difference \( d = 4 \).
Explicit Equation
The explicit formula for an arithmetic sequence is given by:
\[ a_n = a_1 + (n - 1) \cdot d \]
Substituting the values we have:
\[ a_n = 2 + (n - 1) \cdot 4 \]
This simplifies to:
\[ a_n = 2 + 4(n - 1) = 2 + 4n - 4 = 4n - 2 \]
So, the explicit equation is:
\[ a_n = 4n - 2 \]
Recursive Equation
The recursive formula for an arithmetic sequence can be expressed as:
\[ a_n = a_{n-1} + d \]
We also need to specify the first term:
\[ a_1 = 2 \]
So, the recursive equation is:
\[ \begin{align*} a_1 &= 2 \ a_n &= a_{n-1} + 4 \quad \text{for } n > 1 \end{align*} \]
Finding \( a_8 \)
Now, we will find the 8th term \( a_8 \) using the explicit formula:
\[ a_8 = 4(8) - 2 = 32 - 2 = 30 \]
Thus, the 8th term \( a_8 \) is:
\[ \boxed{30} \]