Problem:** A sequence is defined by \( a_1 = 2 \) and \( a_{n+1} = 3a_n + 4 \). What is the value of \( a_4 \)?

["Problem:\nA sequence is defined by ( a_1 = 2 ) and the recurrence relation ( a_{n+1} = 3a_n + 4 ). What is the value of ( a_4 )?", "---", "### Solving the Recurrence: Finding ( a_4 ) in a Linear Nonhomogeneous Sequence", "When working with sequences defined recursively, especially those of the form ( a_{n+1} = r a_n + c ), it’s essential to identify a pattern or closed-form expression to efficiently compute terms without iterative calculation. Here, we are given:\n- Initial value: ( a_1 = 2 )\n- Recurrence: ( a_{n+1} = 3a_n + 4 )", "This is a nonhomogeneous linear recurrence relation. To find ( a_4 ), we can compute the first few terms step-by-step—or derive a general formula. However, computing iteratively is often faster and reliable, especially for small ( n ).", "---", "### Step-by-Step Iterative Calculation", "Start with ( a_1 = 2 ).", "1. Compute ( a_2 ):\n [\n a_2 = 3a_1 + 4 = 3 \ imes 2 + 4 = 6 + 4 = 10\n ]", "2. Compute ( a_3 ):\n [\n a_3 = 3a_2 + 4 = 3 \ imes 10 + 4 = 30 + 4 = 34\n ]", "3. Compute ( a_4 ):\n [\n a_4 = 3a_3 + 4 = 3 \ imes 34 + 4 = 102 + 4 = 106\n ]", "---", "### Verification via Closed-Forms (Optional Insight)", "For deeper understanding, such recurrences can be solved explicitly using characteristic equations and particular solutions. The general solution takes the form:\n[\na_n = A \cdot 3^n + B\n]\nPlugging into the recurrence and solving coefficients yields ( a_n = 4 \cdot 3^{n-1} - 2 ). Testing:\n- ( a_1 = 4 \cdot 3^0 - 2 = 4 - 2 = 2 ) ✅\n- ( a_2 = 4 \cdot 3^1 - 2 = 12 - 2 = 10 ) ✅\n- ( a_3 = 4 \cdot 9 - 2 = 36 - 2 = 34 ) ✅\n- ( a_4 = 4 \cdot 27 - 2 = 108 - 2 = 106 ) ✅", "This confirms our iterative result.", "---", "### Answer:\nThe value of ( a_4 ) is ( \mathbf{106} ).", "---", "### Why Understanding ( a_n ) Matters", "Solving recurrence relations models real-world phenomena like population growth, compound interest, and algorithmic complexity. This problem demonstrates how linear recurrences with constant coefficients increase rapidly—here multiplying by 3 each step—making closed-form solutions essential for efficient computation.", "Knowing how to compute ( a_4 ), ( a_5 ), or general terms empowers better analysis in discrete mathematics and computer science applications.", "---", "### Key Takeaways", "- Start with the base term and apply the recurrence consistently.\n- Verification via direct computation builds confidence.\n- Explicit formulas provide insight beyond single terms.\n- Mastery of sequences helps in modeling and algorithm design.", "---", "Follow-up question: Want to explore how this recurrence behaves as ( n \ o \infty )? The recurrence ( a_{n+1} = 3a_n + 4 ) grows exponentially since the coefficient 3 > 1—so ( a_n ) diverges to infinity.", "---", "Keywords: recurrence relation solution, arithmetic sequence recurrence, ( a_4 ) calculation, nonhomogeneous recurrence, closed-form formula, discrete mathematics, iterative sequence computation."]