Le \(n\)-ème terme est donné par \(a_n = ar^{n-1}\).

["# Understanding the (n)-th Term of a Geometric Sequence: Formula and Applications", "If you’re diving into the world of sequences and series, one of the most essential concepts to master is the (n)-th term formula for geometric sequences. Whether you're calculating terms in finance, science, or mathematics, knowing how to determine the (n)-th term using the formula (a_n = ar^{n-1}) is crucial. In this article, we’ll explore what this formula means, how to use it, and its real-world applications — all optimized for search engines to help you rank well and inform your readers.", "## What is a Geometric Sequence?", "A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant ratio. This ratio is written as (r), and the first term is denoted by (a). Because of this multiplicative pattern, geometric sequences are fundamental in modeling exponential growth and decay.", "## The Formula: (a_n = ar^{n-1}) — Breakdown and Meaning", "The formula for the (n)-th term of a geometric sequence is:", "[\na_n = ar^{n-1}\n]", "Where:\n- (a) = first term ((n = 1))\n- (r) = common ratio (the factor between consecutive terms)\n- (n) = term number (a positive integer)\n- (a_n) = (n)-th term", "This formula allows you to directly calculate any term without listing all prior values — a huge time-saver in large sequences.", "### Why Use (n-1) in the Exponent?\nThe exponent (n-1) ensures you count the number of multiplication steps from the first term. For example:\n- The 1st term ((n=1)) is (ar^{0} = a) — no multiplication needed.\n- The 2nd term ((n=2)) is (ar^{1} = ar).\n- The 3rd term ((n=3)) is (ar^{2} = ar^2).", "This pattern holds for every positive integer (n).", "## How to Use the Formula in Practice", "Let’s walk through a step-by-step example to see the formula in action.", "### Example 1: Given Values\nSuppose (a = 3) and (r = 2). Find the 5th term.", "Using the formula:\n[\na_5 = 3 \ imes 2^{5-1} = 3 \ imes 2^4 = 3 \ imes 16 = 48\n]", "So, the 5th term is 48.", "### Example 2: Real-World Financial Modeling\nGeometric sequences are widely used in financing, interest calculations, and investment growth. For instance, compound interest follows this pattern. If you invest (P) dollars at interest rate (r) compounded annually, the amount after (n) years is:\n[\nA_n = P r^{n-1}\n]\nThus, the term numbering matches perfectly — each year corresponds to an increase by a factor of (r).", "## Step-by-Step Guide to Using (a_n = ar^{n-1})", "1. Identify the first term (a))\n2. Determine the common ratio (r)\n3. Choose the term number (n)\n4. Substitute into the formula: (a_n = ar^{n-1})\n5. Evaluate the expression to find (a_n)", "Always verify your exponent — remember, it’s (n-1), not just (n).", "## Common Mistakes to Avoid", "- Forgetting to subtract 1 from (n) in the exponent: (r^n) is not the same as (r^{n-1}).\n- Confusing (a_n) with the (n)-th summation (series) of the sequence — the formula applies only to individual terms.\n- Misidentifying (a) or (r) due to sequence symmetry — double-check your starting values.", "## Expanding Applications", "Beyond math classrooms, the (ar^{n-1}) formula appears in:\n- Population modeling: Bacterial growth, voter turnout trends.\n- Physics: Signal attenuation in sound or light across distances.\n- Computer science: Algorithm complexity involving repeated halving or doubling.", "## Final Thoughts", "Mastering the (n)-th term formula (a_n = ar^{n-1}) is not just about memorizing an expression — it’s about unlocking a powerful tool for analyzing patterns that grow or shrink exponentially. Whether you’re a student, educator, or professional, using this formula correctly ensures accuracy and confidence in solving sequence-related problems.", "Next time you encounter a geometric sequence, recall: start with (a), multiply by (r) repeatedly, and use the exponent (n-1) to land directly on the (n)-th term.", "Master this formula — master the patterns.", "---", "Keywords for SEO optimization:\n- Geometric sequence formula (a_n = ar^{n-1})\n- How to find (n)-th term\n- Geometric sequence program\n- Exponential growth formula\n- Geometric series and sequences\n- Math formula explanation (a_n = ar^{n-1})", "Meta Description (for search engines):\nLearn the exact formula for the (n)-th term of a geometric sequence: (a_n = ar^{n-1}). Understand how to use it, spot common errors, and apply it to real-world problems like finance, computing, and natural growth patterns.", "Rich Snippet Suggestion:\nUse schema markup for a formula snippet highlighting (a_n = ar^{n-1}) with usage example and key terms.", "---", "Start fortifying your understanding of sequences today — with the formula (a_n = ar^{n-1}), exponential patterns become simple, predictable, and profoundly useful."]