A geometric sequence has first term 3 and common ratio 2. What is the 6th term?
["Understanding a Geometric Sequence: Finding the 6th Term with First Term 3 and Common Ratio 2", "A geometric sequence is a fascinating mathematical pattern where each term after the first is found by multiplying the previous term by a constant called the common ratio. This concept shows up in many real-world applications—from population growth to financial calculations—making it a key topic in algebra and sequence analysis.", "In this article, we’ll explore how to identify and compute the 6th term of a geometric sequence with a clear first term and a defined common ratio. Specifically, we’ll examine the sequence where the first term is 3 and the common ratio is 2.", "### What Defines a Geometric Sequence?", "A geometric sequence follows this simple rule:", "[\na_n = a_1 \cdot r^{(n-1)}\n]", "Where:\n- (a_n) = the (n)-th term\n- (a_1) = the first term\n- (r) = the common ratio\n- (n) = the term number (a positive integer)", "### Given Values for Our Sequence", "For the sequence we’re studying:\n- First term ((a_1)) = 3\n- Common ratio ((r)) = 2\n- Term to find = 6th term ((n = 6))", "### Step-by-Step Calculation of the 6th Term", "Using the formula:", "[\na_6 = 3 \cdot 2^{(6-1)} = 3 \cdot 2^5\n]", "Now compute (2^5):", "[\n2^5 = 32\n]", "Then multiply:", "[\na_6 = 3 \cdot 32 = 96\n]", "### Result", "The 6th term of the geometric sequence with first term 3 and common ratio 2 is 96.", "### Why This Matters", "Recognizing patterns in sequences like this strengthens foundational algebra skills. Problems like finding a specific term help develop logical thinking and problem-solving abilities useful in advanced mathematics, computer science, and engineering.", "### Summary", "- A geometric sequence grows multiplicatively using a fixed common ratio.\n- With (a_1 = 3) and (r = 2), the 6th term is (3 \ imes 2^5 = 96).\n- Understanding these sequences enables smoother progression into more complex mathematical concepts.", "If you're studying sequences and series, practicing with geometric sequences—especially by calculating specific terms—is an effective way to build confidence and precision in algebra.", "---", "You’ve learned how to calculate the 6th term of a geometric sequence with given values. Keep practicing—mastering sequences will pay off in every math journey!"]
