The \( n \)-th term of a geometric sequence is \( a_n = a_1 r^{n-1} \).

["### Understanding The ( n )-th Term of a Geometric Sequence: Formula, Examples, and Applications", "A geometric sequence is one of the fundamental concepts in algebra that describes a pattern of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. Mastering the ( n )-th term formula — ( a_n = a_1 r^{n-1} ) — is essential for solving problems in mathematics, finance, science, and beyond. In this SEO-optimized article, we’ll explore what this formula means, how to use it, and why it’s important.", "---", "### What Is a Geometric Sequence?", "A geometric sequence begins with an initial term ( a_1 ), and every subsequent term is generated by multiplying the prior term by a fixed number ( r ), known as the common ratio. For example, if ( a_1 = 3 ) and ( r = 2 ), the sequence unfolds as:", "[\na_1 = 3,\quad a_2 = 3 \ imes 2 = 6,\quad a_3 = 6 \ imes 2 = 12,\quad a_4 = 12 \ imes 2 = 24,\ldots\n]", "So, the pattern is clear: each term scales the previous one by the ratio ( r ).", "---", "### The ( n )-th Term Formula: ( a_n = a_1 r^{n - 1} )", "The ( n )-th term formula for a geometric sequence gives the value of any term directly, given the first term and the common ratio. Here’s how it works:", "- ( a_1 ): First term (often given or measured)\n- ( r ): Common ratio (ratio between any two consecutive terms)\n- ( n ): Term position (your desired index, starting from 1)\n- Exponent ( n-1 ): Indicates how many times the ratio is applied starting from ( a_1 )", "This formula works for any positive integer ( n ) and avoids the need to compute all previous terms — perfect for predictive modeling.", "---", "### Step-by-Step Example Using the Formula", "Suppose you want to find the 7th term of a sequence where ( a_1 = 5 ) and ( r = 3 ).", "Step 1: Identify parameters\n[\na_1 = 5,\quad r = 3,\quad n = 7\n]", "Step 2: Plug into the formula\n[\na_7 = a_1 r^{n-1} = 5 \ imes 3^{7-1} = 5 \ imes 3^6\n]", "Step 3: Calculate\n[\n3^6 = 729 \quad \Rightarrow \quad a_7 = 5 \ imes 729 = 3645\n]", "So, the 7th term is 3645.", "---", "### Applications of the ( n )-th Term Formula", "The geometric sequence formula is widely applicable in diverse fields:", "- Finance: Calculating compound interest where money grows by a fixed percentage each period\n- Biology: Modeling bacterial growth under ideal conditions (doubling every hour)\n- Physics: Describing exponential decay (e.g., radioactive materials)\n- Computer Science: Analyzing algorithm complexity in geometric growth patterns", "Understanding ( a_n = a_1 r^{n-1} ) empowers learners to analyze and predict such growth scenarios with precision.", "---", "### Tips for Using the Formula", "- Confirm the ratio: Ensure ( r = \frac{a_{n+1}}{a_n} ) for all ( n ) in the sequence.\n- Handle decimals with care: When ( r < 1 ), terms shrink; if ( r > 1 ), they grow exponentially.\n- Remember the exponent trend: always use ( n-1 ), not ( n ).\n- Apply logarithms if solving for unknown ( n ) when given ( a_n ).", "---", "### Summary", "The ( n )-th term of a geometric sequence, defined by ( a_n = a_1 r^{n-1} ), is a cornerstone tool in algebra. It encapsulates exponential growth or decay in a concise formula, enabling quick computation and insightful modeling. Whether you’re solving homework, analyzing investment returns, or simulating natural processes, mastering this formula unlocks powerful problem-solving skills.", "Keywords: geometric sequence, ( a_n = a_1 r^{n-1} ), formula explanation, exponential growth, algebra, compound interest, mathematics education.", "---", "### References", "- Khan Academy. (n.d.). Geometric sequences and series.\n- Paul’s Online Math Notes. (2023). Geometric Sequences and Series.\n- NumDialog. (2024). Geometric Sequence Formula and Applications.", "---", "Optimized for search engines with relevant keywords, concise explanations, and practical relevance, this article educates readers and boosts visibility for queries about geometric sequences and their key formula."]