A geometric sequence has a first term of 3 and a common ratio of 2. What is the 6th term?
["Understanding a Geometric Sequence: Finding the 6th Term When the First Term Is 3 and the Common Ratio Is 2", "A geometric sequence is a special kind of mathematical pattern where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This concept is fundamental in algebra and appears in diverse fields—from finance to science—due to its fast growth nature.", "### What Is a Geometric Sequence?", "A geometric sequence follows the rule:\n[\na_n = a_1 \ imes r^{(n-1)}\n]\nWhere:\n- ( a_n ) = the ( n )-th term\n- ( a_1 ) = the first term\n- ( r ) = the common ratio\n- ( n ) = term position (a positive integer)", "### Given Values in This Problem", "In our specific sequence:\n- The first term ( a_1 = 3 )\n- The common ratio ( r = 2 )\n- We want to find the 6th term, so ( n = 6 )", "### Applying the Formula", "Substitute the known values into the formula:\n[\na_6 = 3 \ imes 2^{(6-1)} = 3 \ imes 2^5\n]", "Now calculate:\n[\n2^5 = 32\n]\n[\na_6 = 3 \ imes 32 = 96\n]", "### Conclusion", "The 6th term in this geometric sequence is 96. Understanding how to calculate such terms helps in more complex pattern recognition and problem-solving across mathematics and real-world applications.", "Whether you're studying for exams or exploring sequences in logic puzzles, mastering geometric progressions like this one builds a strong foundation in algebra and analytical thinking.", "---", "Key takeaway: In a geometric sequence with first term 3 and a common ratio of 2, the 6th term is 96. Use the formula ( a_n = a_1 \ imes r^{n-1} ) anytime you need to find any term efficiently."]
