The general formula for the \( n \)-th term of an arithmetic sequence is:

["The General Formula for the ( n )-Th Term of an Arithmetic Sequence", "An arithmetic sequence is one of the most fundamental and widely used concepts in mathematics, especially in algebra and number theory. Whether you’re solving real-world problems, analyzing patterns, or studying infinite sequences, understanding the formula for the ( n )-th term of an arithmetic sequence is essential. This article explores the clarity, structure, and application of the general formula:", "[\na_n = a_1 + (n - 1)d\n]", "---", "### What Is an Arithmetic Sequence?", "An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference, usually denoted as ( d ). For example, in the sequence\n[ 3, 7, 11, 15, 19, \dots ]\nthe first term is ( a_1 = 3 ), and each term increases by ( d = 4 ).", "---", "### The General Formula Explained", "The formula for the ( n )-th term (( a_n )) of an arithmetic sequence is:", "[\na_n = a_1 + (n - 1)d\n]", "Where:\n- ( a_n ) = the value of the ( n )-th term\n- ( a_1 ) = the first term of the sequence\n- ( d ) = the common difference between consecutive terms\n- ( n ) = the term number (a positive integer: 1, 2, 3, ...)", "---", "### How It Works", "This formula allows you to determine any term in the sequence without listing all preceding terms. For example, to find the 10th term of an arithmetic sequence with ( a_1 = 5 ) and ( d = 3 ):", "[\na_{10} = 5 + (10 - 1) \cdot 3 = 5 + 27 = 32\n]", "Consequently, both efficient computation and pattern recognition become possible, making this formula indispensable in classrooms, exams, and computational coding.", "---", "### Deriving the Formula: Step-by-Step Insight", "To derive the formula, observe that:", "- The 2nd term: ( a_2 = a_1 + d )\n- The 3rd term: ( a_3 = a_1 + 2d )\n- The ( n )-th term: ( a_n = a_1 + (n - 1)d )", "Each term adds ( d ) more than the previous, and from term 1 onward, there are ( n - 1 ) intervals of size ( d ). Thus, the formula naturally emerges from this logical progression.", "---", "### Real-World Applications", "Understanding this formula extends far beyond textbook problems. It applies in:\n- Finance: Calculating fixed installment payments or depreciation schedules\n- Physics: Modeling linear motion with constant velocity\n- Computer Science: Developing algorithms for iterative processes\n- Data Analysis: Identifying and extrapolating trends in time-series data", "---", "### Why This Formula Matters", "The ( a_n = a_1 + (n - 1)d ) formula is powerful because it encapsulates both simplicity and generality. It transforms sequences into predictable tools for analysis and prediction, offering clear insight into ordered progression governed by a fixed rule.", "Whether you're a student mastering algebra, a teacher explaining patterns, or a programmer implementing mathematical models, grasping this formula is foundational.", "---", "### Summary", "- The general formula for the ( n )-th term of an arithmetic sequence is:\n [\n a_n = a_1 + (n - 1)d\n ]\n- It depends on the first term, common difference, and term number.\n- Enables efficient term computation and trend forecasting.\n- Widely applicable across sciences, engineering, and mathematics.", "Master this formula — it unlocks a world of ordered sequences and linear behavior.", "---", "Ready to apply this formula? Practice with various ( a_1 ) and ( d ), and watch how quickly arithmetic sequences unfold in both theory and real life!"]