The nth term of a geometric sequence is \( a_n = a \cdot r^{n-1} = 2 \cdot 3^{n-1} \).
["The nth Term of a Geometric Sequence: Understanding ( a_n = 2 \cdot 3^{n-1} )", "Understanding the nth term of a geometric sequence is essential in mathematics, especially when modeling exponential growth, financial interest, and scientific phenomena. One commonly encountered form is:", "[\na_n = a \cdot r^{n-1}\n]", "In this article, we explore the specific sequence given by the formula:", "[\na_n = 2 \cdot 3^{n-1}\n]", "where ( a = 2 ) is the first term, ( r = 3 ) is the common ratio, and ( n ) represents the term number in the sequence.", "---", "### 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 fixed, non-zero number called the common ratio, denoted ( r ). The general term of such a sequence is expressed as:", "[\na_n = a \cdot r^{n-1}\n]", "Here, ( a ) is the initial term (when ( n = 1 )), ( r ) governs the growth (or decay) pattern, and ( n ) is the position of the term in the sequence.", "---", "### Breaking Down ( a_n = 2 \cdot 3^{n-1} )", "Given:", "[\na_n = 2 \cdot 3^{n-1}\n]", "- The first term (( a_1 )) occurs when ( n = 1 ):\n [\n a_1 = 2 \cdot 3^{1-1} = 2 \cdot 3^0 = 2 \cdot 1 = 2\n ]", "- The common ratio ( r = 3 ) means each term is 3 times the previous one.", "Let’s examine the first few terms to solidify this understanding:", "- ( a_1 = 2 \cdot 3^{0} = 2 )\n- ( a_2 = 2 \cdot 3^{1} = 6 )\n- ( a_3 = 2 \cdot 3^{2} = 18 )\n- ( a_4 = 2 \cdot 3^{3} = 54 )", "Clearly, each term grows by a factor of 3, consistent with ( r = 3 ).", "---", "### Formula Interpretation", "- Multiplying by 2 initializes the sequence: the first term is 2.\n- Raising 3 to the power ( n-1 ) ensures the exponential growth pattern begins with ( n = 1 ).", "For any positive integer ( n ), plugging it into the formula yields the nth term:", "[\na_n = 2 \cdot 3^{n-1}\n]", "This formula is concise, easy to compute, and ideal for applications in compound growth, such as investments or bacterial growth modeled exponentially.", "---", "### Calculating Any Term", "Using the formula, calculating ( a_n ) becomes straightforward:", "Example: Find the 5th term.", "[\na_5 = 2 \cdot 3^{5-1} = 2 \cdot 3^4 = 2 \cdot 81 = 162\n]", "Or step-by-step:", "- ( a_1 = 2 )\n- ( a_2 = 2 \cdot 3 = 6 )\n- ( a_3 = 6 \cdot 3 = 18 )\n- ( a_4 = 18 \cdot 3 = 54 )\n- ( a_5 = 54 \cdot 3 = 162 )", "Perfectly matching the exponential progression.", "---", "### Real-World Applications", "The formula ( a_n = 2 \cdot 3^{n-1} ) models real-world scenarios where exponential growth or decay is present:", "- Investments: A principal amount doubling in value through geometric progression could follow patterns similar to multiplying by 3, reinvested annually.\n- Population Growth: Certain species, under ideal conditions, may grow by tripling each period.\n- Science: Radioactive decay or signal decay in physics sometimes use geometric series with rational or irrational ratios, including 3.", "---", "### Conclusion", "The nth term of a geometric sequence given by ( a_n = 2 \cdot 3^{n-1} ) clearly shows:", "- A starting value of 2,\n- Multiplied by a constant ratio of 3,\n- Yielding rapid exponential growth.", "Mastering such formulas enables students and professionals to model, predict, and analyze phenomena involving exponential trends efficiently. Whether calculating term values or exploring applications, this precise geometric progression formula remains a powerful foundational tool.", "---", "Keywords: geometric sequence formula, nth term of geometric sequence, ( a_n = 2 \cdot 3^{n-1} ), exponential growth, ratio ( r ), mathematical pattern, sequence calculation, compound interest model, term sequence.", "---", "Note for Learners & Researchers: When working with geometric sequences, always identify ( a ), ( r ), and ( n ) clearly, then plug values systematically. Understanding terms like ( a_n = 2 \cdot 3^{n-1} ) strengthens foundations in algebra, calculus, finance, and data science."]
