A sequence is defined by \(a_1 = 2\) and \(a_{n+1} = 3a_n + 1\). Find \(a_4\).
["Title: How to Determine the Fourth Term of a Recursive Sequence: A Step-by-Step Guide", "When studying sequences in mathematics, understanding recursive definitions is essential. A recursive sequence is defined by an initial term and a rule for generating subsequent terms. In this article, we explore a specific recursive sequence defined by:", "- ( a_1 = 2 )\n- ( a_{n+1} = 3a_n + 1 )", "Specifically, we’ll find the fourth term, ( a_4 ), using this recurrence relation.", "---", "### Understanding the Recursive Formula", "The sequence begins with ( a_1 = 2 ). Each subsequent term is generated by multiplying the previous term by 3 and then adding 1:", "[\na_{n+1} = 3a_n + 1\n]", "This formula allows us to compute each term step-by-step, starting from the given initial value.", "---", "### Calculating ( a_2 )", "Using ( a_1 = 2 ), calculate the next term:", "[\na_2 = 3a_1 + 1 = 3 \cdot 2 + 1 = 6 + 1 = 7\n]", "So, ( a_2 = 7 ).", "---", "### Calculating ( a_3 )", "Now use ( a_2 = 7 ) to find ( a_3 ):", "[\na_3 = 3a_2 + 1 = 3 \cdot 7 + 1 = 21 + 1 = 22\n]", "Thus, ( a_3 = 22 ).", "---", "### Calculating ( a_4 )", "Finally, apply the recurrence formula to ( a_3 = 22 ):", "[\na_4 = 3a_3 + 1 = 3 \cdot 22 + 1 = 66 + 1 = 67\n]", "---", "### Final Answer", "The fourth term in the sequence is:", "[\na_4 = 67\n]", "---", "### Why This Matters in Mathematics and Computer Science", "Recursive sequences like this one illustrate fundamental concepts in algorithm design, dynamic programming, and mathematical modeling. Calculating sequence terms step-by-step helps build intuition for iteration and growth patterns — skills vital in fields ranging from data analysis to artificial intelligence.", "By mastering such sequences, students and lifelong learners gain valuable tools for solving complex problems efficiently.", "---", "Keywords: recursive sequence, ( a_n ) definition, ( a_4 ), mathematical iteration, sequence calculation, recurrence relation, step-by-step guide, mathematical recursion.\nMeta Description: Learn how to compute ( a_4 ) for the recursive sequence ( a_1 = 2 ), ( a_{n+1} = 3a_n + 1 ). Step-by-step breakdown and explanation."]
