The \(n\)-th term of a geometric sequence is given by \(a_n = a \cdot r^{n-1}\), where \(a = 3\) and \(r = 4\). The 6th term is \(3 \times 4^{5} = 3 \times 1024 = 3072\).

["Understanding the (n)-th Term of a Geometric Sequence: Example with (a = 3) and (r = 4)", "When studying sequences in mathematics, geometric sequences hold a special place due to their consistent multiplicative pattern. A geometric sequence is defined by its first term and a common ratio that dictates how each term grows or shrinks. If you’re working with a geometric sequence where the first term (a = 3) and the common ratio (r = 4), you can easily compute any term using the standard formula:\n[\na_n = a \cdot r^{n-1}\n]\nHere, (a_n) represents the (n)-th term, (a) is the initial value, (r) is the ratio between consecutive terms, and (n) is the position of the term in the sequence.", "### The General Formula Explained\nPlugging in the given values, the formula becomes:\n[\na_n = 3 \cdot 4^{n-1}\n]\nThis elegant expression tells us how each term evolves: start with 3, then multiply by 4 for each subsequent step.", "### Calculating the 6th Term\nOne of the most common uses of the formula is finding specific terms—such as the 6th term ((n = 6)).", "Using the formula:\n[\na_6 = 3 \cdot 4^{6-1} = 3 \cdot 4^5\n]\nNow compute (4^5):\n[\n4^5 = 4 \ imes 4 \ imes 4 \ imes 4 \ imes 4 = 1024\n]\nThen:\n[\na_6 = 3 \ imes 1024 = 3072\n]\nSo, the 6th term of this geometric sequence is 3072.", "### Why This Formula Matters\nUnderstanding the (n)-th term of geometric sequences helps in modeling exponential growth, financial compounding, population dynamics, and more. With (a = 3) and (r = 4), even modest sequences grow rapidly—this is the power and beauty of geometric progression.", "### Final Insight\nMastering the formula (a_n = a \cdot r^{n-1}) empowers learners and professionals alike to predict behavior and solve problems efficiently in both academic and real-world contexts. For the sequence with (a = 3) and (r = 4), the 6th term confirms how quickly values escalate: starting from 3 and multiplying by 4, the progression unfolds clearly and predictably.", "---", "Keywords: geometric sequence formula, (a_n = a \cdot r^{n-1}), 6th term geometric sequence, exponential growth, common ratio, first term 3, ratio 4, (a_6 = 3072)", "Explore how this simple formula unlocks powerful insights into sequence patterns—essential knowledge for students, educators, and math enthusiasts."]