The \( n \)-th term of a geometric sequence is given by:

["The ( n )-th Term of a Geometric Sequence: A Comprehensive Guide", "Understanding the ( n )-th term of a geometric sequence is fundamental in algebra and serves as a cornerstone for studying exponential growth and decay. Whether you're a student tackling sequences for the first time or a lifelong learner brushing up on core concepts, knowing how to find and apply the ( n )-th term of a geometric sequence is essential.", "### What is a Geometric Sequence?", "A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio, denoted ( r ).", "In general, a geometric sequence begins with an initial value ( a ) (the first term), and each subsequent term follows:", "[\na,\ a r,\ a r^2,\ a r^3,\ \ldots\n]", "Here,\n- ( a ) = first term (( a <br/>\neq 0 )),\n- ( r ) = common ratio (( r <br/>\neq 0 )),\n- ( n ) = term number, a positive integer.", "### Formula for the ( n )-th Term", "The ( n )-th term of a geometric sequence is given by the formula:", "[\na_n = a \cdot r^{n-1}\n]", "This formula allows us to compute any term directly without needing to calculate every term in between—critical for efficiency in problem-solving.", "---", "### Breaking Down the Formula", "- ( a ): The first term of the sequence sets the starting point.\n- ( r^{n-1} ): The base ( r ) raised to the power of ( n - 1 ) determines how many times the ratio multiplies the first term. Since the sequence starts at ( n = 1 ), we use ( n - 1 ), not ( n ).\n- For example, the 1st term (( n = 1 )):\n [\n a_1 = a \cdot r^{1-1} = a \cdot r^0 = a\n ]\n The 2nd term (( n = 2 )):\n [\n a_2 = a \cdot r^{2-1} = a \cdot r\n ]\n The 3rd term (( n = 3 )):\n [\n a_3 = a \cdot r^{3-1} = a \cdot r^2\n ]\n And so on.", "---", "### Applications of the ( n )-th Term Formula", "Calculating individual terms is straightforward, but the real power of this formula lies in its versatility:", "- Modeling real-world phenomena: Geometric sequences model exponential growth (e.g., population growth, compound interest) and exponential decay (e.g., radioactive decay, depreciation).\n- Finance: Used to compute future value of investments with fixed interest rates.\n- Science and engineering: Critical in understanding rates of change, signal decay, and fractal patterns.\n- Problem-solving: Efficiently solved for unknown terms or initial values in word problems.", "---", "### Example", "Let’s say a geometric sequence starts with ( a = 3 ) and has a common ratio ( r = 2 ). Find the 5th term.", "Using the formula:", "[\na_5 = a \cdot r^{5-1} = 3 \cdot 2^4 = 3 \cdot 16 = 48\n]", "This makes sense:\n- ( a_1 = 3 )\n- ( a_2 = 6 )\n- ( a_3 = 12 )\n- ( a_4 = 24 )\n- ( a_5 = 48 )", "Our formula correctly computes each step.", "---", "### Common Mistakes to Avoid", "- Misidentifying ( n ): Remember ( n ) is the term number, so use ( n - 1 ) in the exponent.\n- Forgetting ( r^0 = 1 ): The first term always equals ( a ), not ( a \cdot r^0 ) unless explicitly using the formula.\n- Using addition instead of exponentiation: Terms grow multiplicatively, not additively—never compute ( ar + ar + ar \cdots ).", "---", "### Summary", "- The ( n )-th term of a geometric sequence is ( a_n = a \cdot r^{n-1} ).\n- This formula derives directly from the definition of geometric sequences using repeated multiplication.\n- It enables efficient calculation of any term without enumerating past values.\n- Widely applicable in science, finance, and advanced mathematics.", "Mastering the ( n )-th term of a geometric sequence unlocks deeper insights into exponential relationships and prepares you for more complex mathematical modeling. Whether solving equations, analyzing growth patterns, or working with financial projections, this formula is your essential tool.", "---", "Further Reading\n- Exponential Functions and Their Graphs\n- Applications of Geometric Sequences in Finance\n- Introduction to Recursive Sequences", "Keywords: geometric sequence, ( n )-th term formula, exponential sequence, common ratio, algebra, math tutorial, exponential growth, math concepts, sequence formula."]