A geometric sequence has first term \(a = 4\) and common ratio \(r = 3\). What is the fifth term?
["### Understanding Geometric Sequences: Finding the Fifth Term with First Term (a = 4) and Common Ratio (r = 3)", "A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This concept is fundamental in mathematics, especially when analyzing exponential growth, patterns, and recursive relationships. In this article, we’ll explore how to calculate any term in a geometric sequence and specifically determine the fifth term when the first term (a = 4) and the common ratio (r = 3).", "#### What Is a Geometric Sequence?", "A geometric sequence follows the formula:\n[\na_n = a \cdot r^{n-1}\n]\nwhere:\n- (a_n) is the (n)-th term,\n- (a) is the first term,\n- (r) is the common ratio,\n- (n) is the term number.", "#### Given Parameters", "- First term (a = 4)\n- Common ratio (r = 3)\n- Term to find: (a_5) (the fifth term)", "#### How to Calculate the Fifth Term", "Using the general term formula:\n[\na_5 = a \cdot r^{5-1} = a \cdot r^4\n]\nSubstitute (a = 4) and (r = 3):\n[\na_5 = 4 \cdot 3^4\n]\nCalculate (3^4):\n[\n3^4 = 3 \ imes 3 \ imes 3 \ imes 3 = 81\n]\nNow multiply:\n[\na_5 = 4 \ imes 81 = 324\n]", "#### Conclusion", "The fifth term in the geometric sequence with first term 4 and common ratio 3 is 324. This straightforward computation illustrates how geometric sequences enable quick prediction of future terms using simple exponentiation.", "Understanding geometric sequences is essential not only for math education but also for real-world applications such as finance, population growth, physics, and computer science—making it a key topic in STEM learning and problem-solving.", "---", "Key Takeaways:\n- Use the formula (a_n = a \cdot r^{n-1}) to find any term in a geometric sequence.\n- With (a = 4) and (r = 3), the fifth term (a_5 = 324).\n- Geometric sequences model exponential patterns and thrive in diverse scientific and financial contexts."]
