The \(n\)-th term of an arithmetic sequence is given by \(a_n = a_1 + (n-1) \times d\).

["# The (n)-th Term Formula in Arithmetic Sequences: Mastering This Fundamental Concept", "Understanding the (n)-th term of an arithmetic sequence is essential for students, educators, and math enthusiasts alike. This fundamental formula unlocks deeper insights into sequences, series, and real-world applications such as predictions, scheduling, and financial planning. In this article, we dive into the expression (a_n = a_1 + (n - 1) \ imes d), explain its components, derive how it works, and explore its practical uses.", "---", "## What Is an Arithmetic Sequence?", "An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference (d) to the previous term. For example, the sequence 3, 7, 11, 15, … is arithmetic because each term increases by (d = 4).", "The sequence is defined by:\n- First term ((a_1)): The starting value.\n- Common difference ((d)): The constant garbage between consecutive terms.", "---", "## Decoding the (n)-th Term Formula", "The (n)-th term of an arithmetic sequence is given by:", "[\na_n = a_1 + (n - 1) \ imes d\n]", "### What does each symbol represent?", "- (a_n): The (n)th term in the sequence.\n- (a_1): The first term (known and fixed).\n- (n): The position of the term (any positive integer).\n- (d): The common difference (constant between terms).", "---", "### Why Subtract 1?", "The formula uses (n - 1) because the first term ((n = 1)) corresponds ((1 - 1) \ imes d = 0), so it equals (a_1) exactly. As (n) increases, the ((n - 1)) factor effectively counts how many intervals of size (d) lie after the first term.", "---", "## Deriving the Formula", "Consider an arithmetic sequence starting with (a_1):", "- First term: (a_1 = a_1 + (1 - 1) \cdot d = a_1)\n- Second term: (a_2 = a_1 + (2 - 1) \cdot d = a_1 + d)\n- Third term: (a_3 = a_1 + (3 - 1) \cdot d = a_1 + 2d)\n- (n)th term: (a_n = a_1 + (n - 1) \cdot d)", "This derivation confirms the formula’s consistency with sequence behavior.", "---", "## Examples That Illustrate the Concept", "### Example 1: Simple Sequence\nGiven (a_1 = 5), (d = 3), find (a_6):\n[\na_6 = 5 + (6 - 1) \ imes 3 = 5 + 15 = 20\n]", "Check:\n5, 8, 11, 14, 17, 20 — confirmed.", "### Example 2: Real-Life Application\nIf a bus route starts at terminal 2 and each subsequent bus arrives 4 minutes later, its (n)-th term is:\n(a_n = 2 + (n - 1) \ imes 4).\nThe 10th bus arrives at:\n(a_{10} = 2 + 9 \ imes 4 = 38) minutes.", "---", "## Step-by-Step Guide to Use the Formula", "1. Identify (a_1): Know the first term.\n2. Determine (d): Compute the difference between any two consecutive terms.\n3. Choose (n): Decide which term’s position you want ((n)).\n4. Plug into the formula:\n[\na_n = a_1 + (n - 1) \ imes d\n]\n5. Calculate: Simplify the expression to get (a_n).", "---", "## Advanced Insights and Practice Problems", "- Finding (a_1): Rearranged formula: (a_1 = a_n - (n - 1)d).\n- Finding (d): Use (d = a_n - a_{n-k} \div k) for multiple terms.\n- Solving for (n): Reverse the formula:\n[\nn = \frac{a_n - a_1}{d} + 1\n]", "### Quick Practice:", "If (a_1 = 12), (d = -2), find (a_5):\n[\na_5 = 12 + (5 - 1) \ imes (-2) = 12 - 8 = 4\n]", "---", "## Real-World Applications", "- Finance: Calculating sequential payments with fixed increments.\n- Manufacturing: Predicting output growth or production costs.\n- Education: Modeling exam progression or course difficulty.\n- Nature: Modeling regular patterns like ring spacing in trees.", "---", "## Conclusion", "The formula (a_n = a_1 + (n - 1) \ imes d) is a cornerstone of arithmetic sequences. By mastering this concept, students and professionals unlock powerful tools for modeling linear growth and recurring patterns. Whether solving textbook problems or analyzing real-world data, understanding how to derive and apply this formula is vital.", "---", "### Key Search Intentions Covered:\n- Definition of (a_n) in arithmetic sequences\n- Step-by-step derivation and explanation\n- Practical examples and real-world uses\n- Problem-solving techniques and reversing the formula", "Keywords: arithmetic sequence formula, (n)-th term, linear sequence, common difference, mathematical formula explained, sequence derivation, real-world applications", "---\nStart calculating your sequence progress today—mastering (a_n = a_1 + (n - 1) \cdot d) is your first step!"]