Question: If \( a_2 = 8 \), what are the first three terms of the sequence defined by the recursive…

If \( a_2 = 8 \), what are the first three terms of the sequence defined by the recursive formula?

\[ a_n = 3a_{n-1} + 2 \]

Options:

• 2, 8, 26
• 5, 8, 26
• 8, 26, 80
• 8, 2, 0

Solution

To find the first three terms of the sequence defined by the recursive formula, we start with: Given: \( a_2 = 8 \) Recursive formula: \( a_n = 3a_{n-1} + 2 \) Step 1: Find \( a_3 \). \[ a_3 = 3a_2 + 2 = 3(8) + 2 = 24 + 2 = 26 \] Step 2: Find \( a_4 \). \[ a_4 = 3a_3 + 2 = 3(26) + 2 = 78 + 2 = 80 \] Thus, the first three terms are \( a_2 = 8 \), \( a_3 = 26 \), \( a_4 = 80 \). So, the correct sequence is \( 8, 26, 80 \).