A sequence is defined recursively by \( a_1 = 2 \) and \( a_{n+1} = 3a_n + 1 \). Find the fifth term, \( a_5 \).

["Understanding Recursive Sequences: Find the Fifth Term ( a_5 ) Defined by ( a_1 = 2 ) and ( a_{n+1} = 3a_n + 1 )", "In mathematics, recursive sequences offer a powerful way to define values iteratively. Here, we explore a specific sequence defined by:", "[\na_1 = 2 \quad \ ext{and} \quad a_{n+1} = 3a_n + 1\n]", "This recursive formula means each term depends on the previous one. In this article, we’ll compute the fifth term, ( a_5 ), step-by-step, illustrating how recursive definitions unfold through successive applications.", "---", "### Step-by-Step Computation of ( a_5 )", "Start with the base case:", "[\na_1 = 2\n]", "Now, apply the recursive rule ( a_{n+1} = 3a_n + 1 ) repeatedly:", "Step 1: Compute ( a_2 )\nUsing ( n = 1 ):\n[\na_2 = 3a_1 + 1 = 3(2) + 1 = 6 + 1 = 7\n]", "Step 2: Compute ( a_3 )\nUsing ( n = 2 ):\n[\na_3 = 3a_2 + 1 = 3(7) + 1 = 21 + 1 = 22\n]", "Step 3: Compute ( a_4 )\nUsing ( n = 3 ):\n[\na_4 = 3a_3 + 1 = 3(22) + 1 = 66 + 1 = 67\n]", "Step 4: Compute ( a_5 )\nUsing ( n = 4 ):\n[\na_5 = 3a_4 + 1 = 3(67) + 1 = 201 + 1 = 202\n]", "---", "### Final Result", "After following the recursive steps carefully, the fifth term is:", "[\n\boxed{a_5 = 202}\n]", "---", "### Why Recursion Matters", "Recursive sequences like this model many real-world phenomena—from population growth to algorithm complexity in computer science. Understanding how to compute each term helps build intuition about patterns and behavior over iterations.", "While direct formulas exist for some linear recursions, problems like this often rely on iteration—especially with non-homogeneous forms like ( a_{n+1} = 3a_n + 1 ), where a pattern emerges only through repeated calculation.", "Memorizing recursive definitions is valuable, but practicing substitution is key to mastering such sequences.", "---", "### Conclusion", "Computing ( a_5 ) using ( a_1 = 2 ) and ( a_{n+1} = 3a_n + 1 ) reveals how recursion builds outcomes step-by-step. The result, ( a_5 = 202 ), confidently follows through each iteration. Whether solving math problems or analyzing algorithmic behavior, mastering recursion enhances logical thinking and problem-solving skills.", "For further practice, attempt computing higher terms or modifying the rule to observe how the sequence evolves.", "---", "Keywords: recursive sequence, ( a_{n+1} = 3a_n + 1 ), compute ( a_5 ), iterative computation, mathematical recursion, finding sequence terms.\nMeta description: Compute the fifth term of a recursive sequence defined by ( a_1 = 2 ) and ( a_{n+1} = 3a_n + 1 ). Step-by-step solution and explanation."]