The \( n \)-th term is given by \( a_n = a + (n-1)d \).

["# Understanding the ( n )-th Term of an Arithmetic Sequence: The Formula That Rules It All", "When studying mathematics—especially algebra—one of the fundamental concepts you’ll encounter is the arithmetic sequence. A sequence is an ordered list of numbers, and in the case of arithmetic sequences, each term follows a consistent, predictable pattern. At the heart of this pattern lies a simple yet powerful formula:", "### The ( n )-th Term Formula: ( a_n = a + (n-1)d )", "This elegant expression allows you to find any term in an arithmetic sequence without needing to list all preceding terms. Whether you're solving problems, analyzing patterns, or working through real-world applications, understanding this formula is essential.", "---", "## What Is an Arithmetic Sequence?", "An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This common difference is denoted by ( d ). For example, in the sequence:\n( 3, 7, 11, 15, 19, \ldots )\nthe first term ( a = 3 ) and the common difference ( d = 4 ).", "Because the difference remains unchanged, you can rely on a simple rule to generate each term.", "---", "## Decoding the ( n )-th Term Formula", "The general formula for the ( n )-th term ( a_n ) of an arithmetic sequence is:", "[\na_n = a + (n - 1)d\n]", "### What do the variables mean?\n- ( a ): The first term of the sequence\n- ( d ): The common difference between consecutive terms\n- ( n ): The position of the term you want to find (e.g., ( n = 1 ) for the first term, ( n = 2 ) for the second term, etc.)", "### Why ( n - 1 )?\nBecause the sequence starts at index ( n = 1 ), the difference ( d ) repeats ( n - 1 ) times to reach the ( n )-th term. For example, to get the 5th term, you add ( d ) four times:\n( a_5 = a + 4d )", "---", "## Examples to Make It Clear", "Let’s apply the formula with a concrete example.", "Example:\nSequence: ( a = 5 ), ( d = 3 )", "Find the 7th term (( n = 7 )).", "Using the formula:\n[\na_7 = 5 + (7 - 1) \ imes 3 = 5 + 6 \ imes 3 = 5 + 18 = 23\n]", "So, the 7th term is ( 23 ).", "---", "## How to Use the Formula in Real Life", "The ( n )-th term formula isn't just for classrooms—it applies to countless real-world scenarios:", "- Finance: Calculating compound interest with constant additions\n- Population Studies: Modeling steady annual growth\n- Engineering: Predicting positions in linear motion\n- Retail: Scheduling fixed discount intervals", "For instance, if a store offers a new customer a $2 discount starting from week 1, the discount sequence is arithmetic:\nWeek 1: $2, Week 2: $4, Week 3: $6, …\nSo, ( a = 2 ), ( d = 2 ), and the 10th week discount is:\n[\na_{10} = 2 + (10 - 1) \ imes 2 = 2 + 18 = 20\n]", "---", "## Tips for Mastering the Formula", "- Identify ( a ) and ( d ) quickly when given a sequence.\n- Remember: ( n ) starts at 1, so ( n - 1 ) determines how many times ( d ) is added.\n- Practice with different values to internalize the pattern.\n- Visual cues like number lines or tables help reinforce understanding.", "---", "## Conclusion", "The formula ( a_n = a + (n-1)d ) is a cornerstone of arithmetic sequences—simple yet profoundly useful. By mastering this expression, you gain a powerful tool to analyze linear growth, solve timing problems, and unlock deeper mathematical insights. Whether you're a student mastering algebra or a professional applying mathematical models, understanding the ( n )-th term empowers you to predict and interpret trends with confidence.", "Start with small examples, practice consistently, and soon this formula will become an intuitive part of your mathematical toolkit.", "---", "Keywords: ( n )-th term formula, arithmetic sequence, common difference ( d ), algebra, sequence generation, mathematical formula, linear growth, real-world applications."]