A geometric sequence starts with 3 and has a common ratio of 2. What is the 6th term of the sequence?
["Understanding a Geometric Sequence: Starting with 3, Common Ratio of 2, and Finding the 6th Term", "When exploring sequences in mathematics, geometric sequences are among the most fascinating and widely used patterns. A geometric sequence begins with an initial term and multiplies by a fixed value—known as the common ratio—for each consecutive term. In this article, we’ll dive into a specific geometric sequence that starts with 3 and has a common ratio of 2, and we’ll calculate the 6th term to help deepen your understanding of how these sequences work.", "### What Is a Geometric Sequence?", "A geometric sequence is defined by its first term (a_1) and a common ratio (r), where each subsequent term is found by multiplying the previous term by (r):", "[\na_n = a_1 \ imes r^{(n-1)}\n]", "Here:\n- (a_n) is the (n)th term\n- (a_1) is the first term\n- (r) is the common ratio\n- (n) is the term position", "### Our Specific Sequence", "For the sequence given:\n- First term (a_1 = 3)\n- Common ratio (r = 2)", "To find any term in the sequence, use the formula:", "[\na_n = 3 \ imes 2^{(n-1)}\n]", "### Calculating the 6th Term ((a_6))", "Let’s substitute (n = 6) into the formula:", "[\na_6 = 3 \ imes 2^{(6-1)} = 3 \ imes 2^5\n]", "Now calculate (2^5 = 32):", "[\na_6 = 3 \ imes 32 = 96\n]", "So, the 6th term of the geometric sequence starting with 3 and a common ratio of 2 is 96.", "### Why This Matters", "Understanding geometric sequences is crucial in many areas—from finance and population growth models to computer algorithms and physics. Recognizing patterns like this empowers learners to solve real-world problems involving exponential change efficiently.", "### Summary", "- Initial term: 3\n- Common ratio: 2\n- 6th term: (a_6 = 3 \ imes 2^{5} = 96)", "This geometric sequence grows rapidly due to its doubling pattern: 3 → 6 → 12 → 24 → 48 → 96 (the 6th term). Mastering geometric sequences makes investigating patterns in numbers both accessible and exciting.", "---", "Try it yourself: Use the formula (a_n = 3 \ imes 2^{(n-1)}) to find other terms and explore how doubling each time shapes exponential growth!", "---", "Keywords: geometric sequence, common ratio, 6th term, exponential growth, math formula, educational example\nMeta description: Learn how a geometric sequence starting at 3 with a ratio of 2 yields a 6th term of 96 using the formula and step-by-step calculation."]
