Which recursive formula can be used to represent the sequence [tex]$2, 6, 10, 14, 18, \ldots$[/tex]?

A. [tex]a_n = a_{n-1} - 4[/tex]

B.
[tex]\begin{array}{l}
a_1 = 2 \\
a_n = a_{n-1} + 4
\end{array}[/tex]

C.
[tex]\begin{array}{l}
a_1 = 0 \\
a_n = a_{n-1} + 4
\end{array}[/tex]

D.
[tex]\begin{array}{l}
a_1 = 1 \\
a_n = a_{n-1} + 4
\end{array}[/tex]



Answer :

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]