A sequence is defined such that the first term is 3, and each subsequent term is twice the previous term plus 1. What is the 5th term in this sequence?
["Understanding a Recursive Sequence: Starting with 3 and Defined by a Recurrence Relation", "In mathematics, sequences provide a clear and structured way to explore patterns and relationships. One particularly interesting type is a recursive sequence, where each term is defined based on the previous one. Today, we explore a sequence defined with a simple yet powerful rule: the first term is 3, and each subsequent term is twice the previous term plus 1. This article breaks down how this sequence unfolds, demonstrates the first five terms, and reveals how to calculate the fifth term — making it easy to understand and apply this concept.", "---", "### What Is the Sequence Defined By?", "The sequence starts with:\n- First term (a₁) = 3\n- Recursive rule: For n ≥ 2, each term is defined as:\n [\n a_n = 2a_{n-1} + 1\n ]", "This recurrence relation means that to get the next term, simply take the prior term, multiply by 2, then add 1.", "---", "### Calculating the First Five Terms", "Let’s compute each term step by step using the recurrence:", "1. First term (a₁):\n [\n a_1 = 3\n ]", "2. Second term (a₂):\n [\n a_2 = 2 \ imes a_1 + 1 = 2 \ imes 3 + 1 = 6 + 1 = 7\n ]", "3. Third term (a₃):\n [\n a_3 = 2 \ imes a_2 + 1 = 2 \ imes 7 + 1 = 14 + 1 = 15\n ]", "4. Fourth term (a₄):\n [\n a_4 = 2 \ imes a_3 + 1 = 2 \ imes 15 + 1 = 30 + 1 = 31\n ]", "5. Fifth term (a₅):\n [\n a_5 = 2 \ imes a_4 + 1 = 2 \ imes 31 + 1 = 62 + 1 = 63\n ]", "---", "### Summary of the First Five Terms", "| Term (n) | Value |\n|----------|-------------|\n| a₁ | 3 |\n| a₂ | 7 |\n| a₃ | 15 |\n| a₄ | 31 |\n| a₅ | 63 |", "---", "### What Is the 5th Term?", "The 5th term in this sequence is 63. This result follows naturally from the recursive formula, illustrating how simple rules can generate increasingly complex values.", "---", "### Why This Sequence Matters", "Understanding recursive sequences helps build analytical thinking useful in computer science, algebra, and algorithm design. This particular sequence demonstrates exponential growth tempered by a constant addition — a common pattern in real-world models involving compound interest, population growth, and algorithmic complexity.", "---", "### Final Thoughts", "By following a straightforward rule — doubling and adding one — we discovered a rapidly growing sequence with a clear fifth term. Whether solving math problems, coding algorithms, or analyzing patterns, recognizing such recursive definitions empowers deeper insight and problem-solving flexibility.", "So, if your challenge is: first term is 3, each next term is double the prior plus one, the 5th term is 63 — simple to compute, powerful to understand.", "---", "Key Terms for SEO:\nrecursive sequence, defining sequence, first term 3, each subsequent term doubles plus 1, fifth term calculation, exponential growth sequence, mathematical recursion example, sequence problem solving."]
