Let's carefully analyze the sequence [tex]\(2, 5, 8, 11, 14, \ldots\)[/tex] and determine the correct recursive formula that represents it.
1. Identify the pattern:
- The first term is [tex]\(2\)[/tex].
- The second term is [tex]\(5\)[/tex], which is [tex]\(2 + 3\)[/tex].
- The third term is [tex]\(8\)[/tex], which is [tex]\(5 + 3\)[/tex].
- The fourth term is [tex]\(11\)[/tex], which is [tex]\(8 + 3\)[/tex].
- The fifth term is [tex]\(14\)[/tex], which is [tex]\(11 + 3\)[/tex].
From this, we can see that each term is obtained by adding [tex]\(3\)[/tex] to the previous term.
2. Formulate the recursive relationship:
- Let's denote the [tex]\(n\)[/tex]-th term by [tex]\(a_n\)[/tex].
- The first term, [tex]\(a_1\)[/tex], is [tex]\(2\)[/tex].
- For [tex]\(n \geq 2\)[/tex], each term [tex]\(a_n\)[/tex] is [tex]\(a_{n-1} + 3\)[/tex].
3. Evaluate the given options:
- Option A: [tex]\(a_n = a_{n-1} \cdot 3\)[/tex]
- This option suggests that each term is obtained by multiplying the previous term by [tex]\(3\)[/tex]. For example, [tex]\(a_2\)[/tex] would be [tex]\(2 \cdot 3 = 6\)[/tex]. This does not match our sequence.
- Incorrect.
- Option B: [tex]\( \begin{array}{l} a_1 = 2 \\ a_n = a_{n-1} + 3 \end{array} \)[/tex]
- This matches our observation exactly. The first term [tex]\(a_1\)[/tex] is [tex]\(2\)[/tex] and each subsequent term is obtained by adding [tex]\(3\)[/tex] to the previous term.
- This is correct.
- Option C: [tex]\(a_n = a_{n-1} \cdot 3\)[/tex]
- This is the same as Option A and similarly incorrect.
- Option D: [tex]\( \begin{array}{l} a_1 = 0 \\ a_n = a_{n-1} + 3 \end{array} \)[/tex]
- This suggests starting with [tex]\(0\)[/tex] and adding [tex]\(3\)[/tex] each time. The first term here would be [tex]\(0\)[/tex] which is not the start of our sequence.
- Incorrect.
Therefore, the recursive formula that correctly represents the sequence [tex]\(2, 5, 8, 11, 14, \ldots\)[/tex] is:
[tex]\[ \boxed{\text{B}} \qquad \begin{array}{l} a_1 = 2 \\ a_n = a_{n-1} + 3 \end{array} \][/tex]
