A sequence is defined recursively by \( a_1 = 2 \) and \( a_{n} = 3a_{n-1} + 4 \) for \( n \geq 2 \). What is \( a_4 \)?

["### Unlocking Recursive Sequences: How to Find ( a_4 ) When ( a_1 = 2 ) and ( a_n = 3a_{n-1} + 4 )", "Understanding recursive sequences is a foundational concept in mathematics and computer science. One powerful way to analyze such sequences is by computing each term step-by-step using the recursive rule. Today, we’ll explore how to determine the value of ( a_4 ) in the sequence defined by:", "- Base case: ( a_1 = 2 )\n- Recursive formula: ( a_n = 3a_{n-1} + 4 ) for ( n \geq 2 )", "Let’s walk through the sequence step by step to find ( a_4 ).", "---", "### Step 1: Calculate ( a_2 )", "Start with the base term and apply the recursive definition:", "[\na_2 = 3a_1 + 4 = 3 \ imes 2 + 4 = 6 + 4 = 10\n]", "---", "### Step 2: Calculate ( a_3 )", "Use ( a_2 = 10 ) to compute the next term:", "[\na_3 = 3a_2 + 4 = 3 \ imes 10 + 4 = 30 + 4 = 34\n]", "---", "### Step 3: Calculate ( a_4 )", "Now use ( a_3 = 34 ) to find ( a_4 ):", "[\na_4 = 3a_3 + 4 = 3 \ imes 34 + 4 = 102 + 4 = 106\n]", "---", "### Final Answer", "Thus, the fourth term in the sequence is:", "[\n\boxed{106}\n]", "---", "### Why This Matters", "Recursive sequences like this one appear in algorithms, financial modeling, and computer simulations. By breaking each term down recursively, you gain insight into how values grow exponentially—a key skill in both theoretical and applied mathematics.", "So next time you encounter a recursive definition, remember: knowing how to compute ( a_n ) step-by-step empowers you to solve much larger problems efficiently. Try calculating ( a_5 ) or exploring different initial values to deepen your understanding!"]