A geometric sequence has a first term of 3 and a common ratio of 2. Find the 6th term.
["Understanding Geometric Sequences: Finding the 6th Term When the First Term is 3 and the Common Ratio is 2", "A geometric sequence is a type of mathematical progression where each term after the first is found by multiplying the previous term by a constant called the common ratio. This concept is essential in algebra, finance, science, and many areas of applied mathematics.", "In this article, we’ll explore a specific geometric sequence, learn how to identify its key components, and apply the formula to find the 6th term when:", "- The first term ((a_1)) = 3\n- The common ratio ((r)) = 2", "---", "### What Is a Geometric Sequence?", "A geometric sequence follows the pattern:\n[\na,\ ar,\ ar^2,\ ar^3,\ ar^4,\ \dots\n]\nWhere:\n- (a) = first term\n- (r) = common ratio\n- (n) = term number", "The general formula for the (n)th term is:\n[\na_n = a_1 \ imes r^{(n-1)}\n]", "---", "### Apply the Formula to Find the 6th Term", "Given:\n- (a_1 = 3)\n- (r = 2)\n- We want the 6th term ((a_6)) → (n = 6)", "Substitute into the formula:\n[\na_6 = 3 \ imes 2^{(6-1)} = 3 \ imes 2^5 = 3 \ imes 32 = 96\n]", "---", "### Result and Interpretation", "The 6th term of the sequence is 96.\nThis means, starting from 3 and doubling each time:", "- (a_1 = 3)\n- (a_2 = 3 \ imes 2 = 6)\n- (a_3 = 6 \ imes 2 = 12)\n- (a_4 = 12 \ imes 2 = 24)\n- (a_5 = 24 \ imes 2 = 48)\n- (a_6 = 48 \ imes 2 = 96)", "Checking using the formula confirms the result.", "---", "### Why This Matters", "Understanding how to find any term in a geometric sequence is valuable. Whether calculating compound interest, modeling population growth, or analyzing geometric patterns in art and design, geometric sequences provide clear mathematical tools.", "---", "### Summary", "Given a geometric sequence with first term 3 and common ratio 2:\n- The 6th term is calculated using (a_n = a_1 \ imes r^{(n-1)})\n- (a_6 = 3 \ imes 2^{5} = 96)", "Answer: The 6th term is 96.", "---", "For anyone studying sequences or preparing for exams, mastering geometric progression formulas ensures a strong foundation in algebraic reasoning. Practice with real examples to internalize the logic — it’s simpler than it sounds!"]
