A sequence is defined recursively by \( a_1 = 2 \), \( a_{2} = 3 \), and \( a_{n} = a_{n-1} + 2a_{n-2} \) for \( n \ge 3 \). Find \( a_6 \).

["Understanding Recursive Sequences: Compute ( a_6 ) Using the Recursive Definition", "In mathematics and computer science, sequences defined recursively are powerful tools for modeling patterns and relationships. One such sequence is given by:", "[\na_1 = 2,\quad a_2 = 3,\quad a_n = a_{n-1} + 2a_{n-2} \quad \ ext{for } n \ge 3\n]", "This article explores how to compute the 6th term ( a_6 ) using the recursive definition, demonstrating both step-by-step calculation and insight into recursive sequences.", "### Step 1: Understand the Recursion", "The sequence starts with fixed initial values:\n- ( a_1 = 2 )\n- ( a_2 = 3 )", "For every term beyond the second, the value is defined as the sum of the previous term and twice the term before that:\n- ( a_n = a_{n-1} + 2a_{n-2} )", "This recurrence relation makes the sequence grow at a controlled but accelerating rate.", "### Step 2: Compute the Terms Sequentially", "We compute each term one at a time using the recursive formula.", "- ( a_1 = 2 )\n- ( a_2 = 3 )", "Now calculate ( a_3 ) through ( a_6 ):", "Compute ( a_3 ):\n[\na_3 = a_2 + 2a_1 = 3 + 2(2) = 3 + 4 = 7\n]", "Compute ( a_4 ):\n[\na_4 = a_3 + 2a_2 = 7 + 2(3) = 7 + 6 = 13\n]", "Compute ( a_5 ):\n[\na_5 = a_4 + 2a_3 = 13 + 2(7) = 13 + 14 = 27\n]", "Compute ( a_6 ):\n[\na_6 = a_5 + 2a_4 = 27 + 2(13) = 27 + 26 = 53\n]", "### Final Result", "The sixth term in the sequence is\n[\n\boxed{53}\n]", "### Why Recursive Sequences Matter", "Recursive definitions like this are foundational in algorithm design, especially in dynamic programming, where solving large problems involves breaking them into smaller, structured subproblems. Understanding recursion helps in optimizing computations and detecting patterns that recur across multiple domains—from financial modeling to biological systems.", "### Summary", "Given:\n- ( a_1 = 2 )\n- ( a_2 = 3 )\n- ( a_n = a_{n-1} + 2a_{n-2} ) for ( n \geq 3 )", "We find:\n[\na_6 = 53\n]", "This example highlights how recursion enables sequential computation with clear, repeatable logic—essential for both theoretical study and practical implementation."]