To determine the recursive formula for the sequence [tex]\(2, 6, 10, 14, 18, \ldots\)[/tex], let's break it down step-by-step:
1. Identify the first term [tex]\( a_1 \)[/tex]:
The first term of the sequence is given as [tex]\( a_1 = 2 \)[/tex].
2. Identify the common difference [tex]\( d \)[/tex]:
To determine the common difference, subtract the first term from the second term, the second term from the third term, and so on:
[tex]\[
6 - 2 = 4 \\
10 - 6 = 4 \\
14 - 10 = 4 \\
18 - 14 = 4
\][/tex]
Hence, the common difference [tex]\( d \)[/tex] is [tex]\( 4 \)[/tex].
3. Formulate the recursive formula:
A recursive formula for an arithmetic sequence can be written as:
[tex]\[
a_n = a_{n-1} + d
\][/tex]
where [tex]\( a_{n-1} \)[/tex] is the previous term and [tex]\( d \)[/tex] is the common difference. Here, [tex]\( d = 4 \)[/tex].
Therefore, the recursive formula becomes:
[tex]\[
a_n = a_{n-1} + 4
\][/tex]
4. Combine with the initial term:
Including the initial term, we get:
[tex]\[
a_1 = 2 \\
a_n = a_{n-1} + 4 \quad \text{for } n > 1
\][/tex]
This matches with option B from the given choices:
[tex]\[
\boxed{\begin{array}{l}
a_1 = 2 \\
a_n = a_{n-1} + 4
\end{array}}
\][/tex]
