A geometric sequence starts with 3 and has a common ratio of 4. What is the 6th term of the sequence?
["Understanding Geometric Sequences: The 6th Term of a Sequence Starting with 3 and Having a Common Ratio of 4", "A geometric sequence is a specific type of mathematical sequence where each term after the first is found by multiplying the previous term by a constant value known as the common ratio. This special structure makes geometric sequences extremely useful in fields like finance, biology, physics, and computer science. In this article, we’ll explore what a geometric sequence is, how to calculate its terms, and determine the 6th term when the sequence begins with 3 and uses a common ratio of 4.", "### What Defines a Geometric Sequence?", "A geometric sequence follows this clear pattern:\nIf ( a_1 ) is the first term and ( r ) is the common ratio, then each subsequent term is given by\n[ a_n = a_1 \ imes r^{(n-1)} ]\nHere,\n- ( a_n ) = ( n )th term\n- ( a_1 ) = first term\n- ( r ) = common ratio\n- ( n ) = term position (positive integer)", "### Applying the Formula to the Given Sequence", "For this problem:\n- First term ( a_1 = 3 )\n- Common ratio ( r = 4 )\n- We want the 6th term, so ( n = 6 )", "Plug these values into the formula:\n[ a_6 = 3 \ imes 4^{(6-1)} = 3 \ imes 4^5 ]", "Now calculate ( 4^5 ):\n[ 4^5 = 4 \ imes 4 \ imes 4 \ imes 4 \ imes 4 = 1024 ]\nThen multiply by 3:\n[ a_6 = 3 \ imes 1024 = 3072 ]", "### Conclusion", "The 6th term of the geometric sequence that starts with 3 and has a common ratio of 4 is 3072. Understanding geometric sequences helps unlock powerful tools for modeling exponential growth, compound interest, population studies, and more. Whether you’re a student, educator, or math enthusiast, mastering these concepts brings clearer insight into the patterns shaping the world around us.", "If you’re exploring sequences, remember this formula:\n[ a_n = a_1 \ imes r^{(n-1)} ]\nA simple formula with vast applications!"]
