A sequence is defined recursively as \( a_1 = 2 \), and \( a_{n} = 3a_{n-1} + 4 \) for \( n \geq 2 \). What is the value of \( a_5 \)?

["# Understanding Recursive Sequences: Finding ( a_5 ) in ( a_1 = 2 ) and ( a_n = 3a_{n-1} + 4 )", "Recursive sequences define each term based on the previous one, making them a fascinating concept in mathematics and computer science. Learning how to solve these sequences step-by-step is essential for mastering pattern recognition and algorithmic thinking. In this article, we’ll explore a classic recursive sequence, determine its closed-form formula, and compute the specific value of ( a_5 ).", "## defined recursively: base case and recurrence relation", "The sequence is defined as:", "- Base case: ( a_1 = 2 )\n- Recursive formula: ( a_n = 3a_{n-1} + 4 ) for ( n \geq 2 )", "Such sequences depend on prior values, making direct computation necessary for early terms. However, identifying a pattern—and even a closed-form expression—allows us to compute larger terms efficiently.", "## calculating initial terms step by step", "Let’s compute the first few terms using the recurrence:", "- ( a_1 = 2 )\n- ( a_2 = 3a_1 + 4 = 3(2) + 4 = 6 + 4 = 10 )\n- ( a_3 = 3a_2 + 4 = 3(10) + 4 = 30 + 4 = 34 )\n- ( a_4 = 3a_3 + 4 = 3(34) + 4 = 102 + 4 = 106 )\n- ( a_5 = 3a_4 + 4 = 3(106) + 4 = 318 + 4 = 322 )", "So, ( a_5 = 322 ).", "## finding a closed-form solution (optional but insightful)", "While direct computation gives us ( a_5 = 322 ), solving the recurrence fully reveals deeper structure. The recurrence ( a_n = 3a_{n-1} + 4 ) is linear and nonhomogeneous. Using standard techniques:", "1. Solve the homogeneous part: ( a_n^{(h)} = C \cdot 3^n )\n2. Find a particular solution: Assume a constant ( a_n^{(p)} = A ). Substituting:\n ( A = 3A + 4 \Rightarrow -2A = 4 \Rightarrow A = -2 )\n3. General solution: ( a_n = C \cdot 3^n - 2 )\n4. Use initial condition ( a_1 = 2 ):\n ( 2 = C \cdot 3^1 - 2 \Rightarrow 2 = 3C - 2 \Rightarrow 3C = 4 \Rightarrow C = \frac{4}{3} )\n5. Closed form:\n [\n a_n = \frac{4}{3} \cdot 3^n - 2 = 4 \cdot 3^{n-1} - 2\n ]", "Verify ( a_5 ):\n[\na_5 = 4 \cdot 3^{4} - 2 = 4 \cdot 81 - 2 = 324 - 2 = 322\n]\nConsistent!", "## why knowing ( a_5 ) matters", "Beyond textbook practice, recursive sequences model real-world processes: population growth, compound interest with feedback, branching algorithms, and dynamic programming. Being able to compute specific terms like ( a_5 ) builds fluency in pattern extraction—an essential skill in math, coding, and data science.", "## conclusion", "Recursive definitions smoothly transition from iteration to closed-form insight. For the sequence with ( a_1 = 2 ) and ( a_n = 3a_{n-1} + 4 ), computing ( a_5 ) step-by-step yields 322, and verifying via the derived formula confirms accuracy. Mastery of such sequences empowers problem-solving across disciplines—starting with a simple calculation and expanding into deeper analytical thinking.", "If you're learning sequences, practice computing terms and seek patterns. Understanding recursion isn’t just about finding the next number—it’s about seeing how small rules build complex, predictable patterns.\n( \boxed{322} )"]