A sequence is defined by the recursive formula \( a_1 = 3 \) and \( a_{n+1} = 2a_n + 1 \). What is the 5th term of the sequence?

["### Understanding Recursive Sequences: Solving the Formula ( a_1 = 3 ), ( a_{n+1} = 2a_n + 1 ) and Finding the 5th Term", "In mathematics, sequences defined by recursive formulas are powerful tools for modeling patterns and growth. One classic example is the recursive sequence where the first term is given, and each subsequent term follows a specific rule based on the previous one. The sequence defined by ( a_1 = 3 ) and ( a_{n+1} = 2a_n + 1 ) presents an intriguing recursive pattern that leads to a well-defined progression. In this article, we’ll explore how to compute terms of this sequence step by step, highlight its mathematical behavior, and specifically determine the 5th term.", "---", "#### What Is a Recursive Sequence?", "A recursive sequence defines each term based on prior terms using a recurrence relation. Here, the relation is:", "[\na_1 = 3\n]\n[\na_{n+1} = 2a_n + 1 \quad \ ext{(for } n \geq 1\ ext{)}\n]", "This means each new term doubles the previous one and adds 1.", "---", "#### Calculating the First Few Terms", "Starting from ( a_1 = 3 ), we compute subsequent terms step by step:", "- First term:\n [\n a_1 = 3\n ]", "- Second term:\n Using ( a_2 = 2a_1 + 1 ):\n [\n a_2 = 2 \cdot 3 + 1 = 6 + 1 = 7\n ]", "- Third term:\n [\n a_3 = 2a_2 + 1 = 2 \cdot 7 + 1 = 14 + 1 = 15\n ]", "- Fourth term:\n [\n a_4 = 2a_3 + 1 = 2 \cdot 15 + 1 = 30 + 1 = 31\n ]", "- Fifth term:\n [\n a_5 = 2a_4 + 1 = 2 \cdot 31 + 1 = 62 + 1 = 63\n ]", "---", "#### The Result: The 5th Term Is 63", "Thus, by following the recursive rule consistently, we find:\nThe 5th term of the sequence is ( a_5 = 63 ).", "---", "#### Pattern Insight: Closed-Form Formula", "Besides computing terms recursively, this sequence follows a recognizable closed-form expression. Observing:", "[\na_1 = 3 = 2^1 + 1\n]\n[\na_2 = 7 = 2^2 + 3\n]\n[\na_3 = 15 = 2^3 + 7\n]\n[\na_4 = 31 = 2^4 + 15\n]\n[\na_5 = 63 = 2^5 + 31\n]", "We notice a pattern:\n[\na_n = 2^n + a_{n-1}\n]", "Interestingly, solving this recurrence reveals:\n[\na_n = 2^n + 1 - 2^{n-1} = 2^{n-1} + 1 \cdot 2 + 1 - 2^{n-1} = 2^{n-1} + 1 + (2^{n-1} - 2^{n-1}) + 1 \quad (\ ext{not optimal})\n]", "But testing the closed form ( a_n = 2^{n+1} - 1 ):", "- For ( n = 1 ): ( 2^{2} - 1 = 4 - 1 = 3 ) ✓\n- ( n = 2 ): ( 2^3 - 1 = 8 - 1 = 7 ) ✓\n- ( n = 3 ): ( 2^4 - 1 = 16 - 1 = 15 ) ✓\n- ( n = 4 ): ( 2^5 - 1 = 32 - 1 = 31 ) ✓\n- ( n = 5 ): ( 2^6 - 1 = 64 - 1 = 63 ) ✓", "Indeed, the closed-form solution is:\n[\na_n = 2^{n+1} - 1\n]", "This confirms our recursive calculations and reveals a exponential growth trend—typical of linear nonhomogeneous recursions like this one.", "---", "#### Why This Sequence Matters", "Sequences like this illustrate exponential growth with a constant additive bias, modeling phenomena such as doubling populations with fixed additions. Understanding recursive definitions and computing terms unlocks deeper insight into algorithm design, finance modeling, and computer science recurrence relations.", "---", "### Final Answer:", "The 5th term of the sequence defined by ( a_1 = 3 ) and ( a_{n+1} = 2a_n + 1 ) is ( \boxed{63} ).", "You can compute each term step by step or use the closed form ( a_n = 2^{n+1} - 1 ) to quickly find terms regardless of position. This sequence exemplifies how simple rules generate powerful mathematical behavior."]