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

["# The ( n )-th Term of a Geometric Sequence: A Complete Guide", "Understanding geometric sequences is fundamental in mathematics, especially in algebra and series analysis. One of the most essential concepts is determining the ( n )-th term of a geometric sequence using the formula:", "[\na_n = a_1 \cdot r^{n-1}\n]", "This article explores the meaning, derivation, applications, and step-by-step calculation of the ( n )-th term in geometric sequences, helping students and learners grasp this critical concept with clarity and confidence.", "---", "## What Is a Geometric Sequence?", "A geometric sequence is an ordered list of numbers where each term after the first is found by multiplying the previous term by a constant ratio called ( r ), known as the common ratio.", "For example, in the sequence:\n[\n2,\ 6,\ 18,\ 54,\ \ldots\n]\nwe see that each term is multiplied by 3 to obtain the next:\n[\nr = \frac{6}{2} = 3\n]", "This consistent ratio defines the behavior and structure of geometric sequences, making it easier to extrapolate any term forward.", "---", "## The Formula for the ( n )-th Term", "The general formula for the ( n )-th term (( a_n )) in a geometric sequence is:\n[\na_n = a_1 \cdot r^{n-1}\n]\nwhere:\n- ( a_n ) = value of the ( n )-th term\n- ( a_1 ) = first term of the sequence\n- ( r ) = common ratio\n- ( n ) = term number (a positive integer)", "Why subtract 1 from ( n )?\nBecause the first term (( n = 1 )) uses ( r^0 = 1 ), so the pattern works smoothly from the start. This ensures consistency across every term in the sequence.", "---", "## Step-by-Step: How to Find the ( n )-th Term", "1. Identify ( a_1 ): The initial term in the sequence.\n2. Determine the common ratio ( r ): Divide any term by its predecessor (e.g., ( r = T_2 / T_1 )).\n3. Choose the term number ( n ): The position of the term in the sequence.\n4. Apply the formula: Plug values into ( a_n = a_1 \cdot r^{n-1} ).", "### Example 1:\nGiven: ( a_1 = 4 ), ( r = 2 ), find ( a_5 ).", "Method:\n[\na_5 = 4 \cdot 2^{5-1} = 4 \cdot 2^4 = 4 \cdot 16 = 64\n]\nSo, the 5th term is 64.", "### Example 2:\nGiven: the 3rd term is 18, common ratio ( r = 3 ), find ( a_1 ).", "Solve for ( a_1 ):\n[\n18 = a_1 \cdot 3^{3-1} \Rightarrow a_1 = \frac{18}{3^2} = \frac{18}{9} = 2\n]", "---", "## Applications of the ( n )-th Term Formula", "The formula extends beyond simple sequences—it’s a powerful tool in:\n- Finance: Calculating compound interest (( A = P \cdot (1 + r)^t )).\n- Population Modeling: Estimating growth over discrete time periods.\n- Physics: Analyzing exponential decay (radioactive decay, capacitor discharge).\n- Computer Science: Designing algorithms with exponential time/space complexity.", "---", "## Practical Tips for Mastery", "- Always verify ( r ): A neglected mistake in identifying ( r ) leads to incorrect results.\n- Use exponents wisely: Remember ( r^{n-1} ), not ( r^n ).\n- Apply logarithms when needed: If solving for ( n ), use ( n = \log_r(a_n / a_1) + 1 ).\n- Visualize the sequence: Plotting the terms reveals exponential growth or decay clearly.", "---", "## Common Mistakes to Avoid", "- Confusing ( n = 1 ) vs. ( n = 0 ); remember ( a_1 = a_1 \cdot r^0 ).\n- Forgetting to subtract 1 from ( n ) in the exponent.\n- Misapplying ( r ) when it’s not constant across all terms.", "---", "## Conclusion", "Mastering the formula ( a_n = a_1 \cdot r^{n-1} ) unlocks deeper understanding of exponential patterns in mathematics and real-world systems. Whether calculating next-term values, solving complex equations, or modeling natural phenomena, this formula remains indispensable. Practice with varied examples to build confidence—you’ll soon calculate geometric terms effortlessly!", "---", "Keyword-rich FAQ section:\n- What does ( a_n = a_1 \cdot r^{n-1} ) mean?\nIt defines how to compute any term in a geometric sequence given its first term and common ratio.\n- Where is this formula used?\nIn finance, physics, biology, and computer science for modeling growth or decay.\n- How do I solve for unknowns?\nRearranging the formula using logarithms or algebraic manipulation solves for missing values.", "Optimization: This SEO article strategically uses key terms like geometric sequence formula, ( a_n = a_1 \cdot r^{n-1} ), and common ratio, boosting visibility for students, educators, and self-learners searching for clear, accurate math explanations."]