Solution: The sequence of dosages forms an arithmetic progression with first term $ a = 2 $, common difference $ d = 3 $, and last term $ l = 200 $. Using the formula for the $ n $-th term of an arithmetic sequence:
["Understanding Dosage Patterns: The Hidden Math Behind Health and Wellness Progressions", "Ever wondered how medical regimens or supplement schedules align with precise mathematical patterns? One such fascinating structure is the arithmetic sequence—a sequence where each step increases by a consistent amount. Recent interest in structured dosing — from vitamin routines to chronic condition management—has sparked curiosity about how numbers define daily health practices. From real-world applications to digital trends, understanding these sequences supports informed decisions across the U.S. population.", "---", "### Why It Matters: Does This Sequence Matter in Daily Life?", "In an era where wellness tracking and personalized healthcare guide consumer choices, sequences like arithmetic progressions offer more than abstract math. Actualized dosage plans reinforce consistency, support effective treatment timing, and improve adherence—critical factors in health outcomes. While often invisible, these patterns reflect a growing awareness: structured, predictable routines enhance both physical and mental well-being. Beyond supplements and medications, similar principles guide dose intervals in nutrition, physical therapy, and digital health apps tracking behavioral habits.", "---", "### What Is an Arithmetic Sequence — And How Does It Apply?", "An arithmetic sequence follows a clear formula: \naₙ = a + (n – 1) × d \nWhere: \n- a = starting value \n- d = consistent difference between terms \n- n = position in the sequence", "For the well-documented case of dosages: starting at $ a = 2 $, increasing by $ d = 3 $, and culminating at $ l = 200 $, the full list unfolds naturally: 2, 5, 8, 11, ..., up to 200. This mathematical structure reveals predictable, manageable intervals ideal for long-term health planning. Users and providers alike benefit from visualizing clear, calculable intervals—making it easier to adhere to treatment goals without confusion or miscalculation.", "---", "### How Physical Dosage Sequences Work in Practice", "Consider medical or wellness regimens using structured dosage schedules. Patients often follow protocols designed to build tolerance gradually or maximize absorption. The arithmetic sequence ensures steady increments, reducing risk of side effects while allowing the body to adjust. Digital wellness platforms even mirror this logic, suggesting consistent daily check-ins or gradual adjustments in activity and rest. While the examples are non-explicit, the pattern underscores a shift toward smarter, evidence-based personal routines.", "---", "### Common Questions Readers Want to Understand", "How many steps does the sequence include? \nUsing the formula for the $ n $-th term: \n$ 200 = 2 + (n – 1) \ imes 3 $ \nSolving gives $ n = 67 $. There are 67 distinct dosage intervals in this sequence.", "**Can this apply beyond supplements"]
