Question

1. represent the sequence {7, 9, 11, 13, ...} in both recursive and explicit form.

Explanation:

Step1: Identify Sequence Type

The sequence \( \{7, 9, 11, 13, \dots\} \) is arithmetic. Common difference \( d = 9 - 7 = 2 \), first term \( a_1 = 7 \).

Step2: Recursive Formula

For arithmetic sequence, \( a_n = a_{n - 1} + d \), with \( a_1 = 7 \). So recursive: \( a_1 = 7 \), \( a_n = a_{n - 1} + 2 \) for \( n \geq 2 \).

Step3: Explicit Formula

Explicit formula for arithmetic sequence: \( a_n = a_1 + (n - 1)d \). Substitute \( a_1 = 7 \), \( d = 2 \):
\( a_n = 7 + (n - 1) \cdot 2 = 2n + 5 \).

Answer:

• Recursive form: \( a_1 = 7 \), \( a_n = a_{n - 1} + 2 \) (for \( n \geq 2 \))
• Explicit form: \( a_n = 2n + 5 \)