Question

Write an explicit equation and recursive equation for each arithmetic sequence. then find the indicated term.
a_(1) =2, d = 4, a_(8)

Answer 1

To find the explicit and recursive equations for the given arithmetic sequence, we'll start with the information provided:

1. The first term \( a_1 = 2 \).
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:

\[
\boxed{30}
\]