A geometric sequence has a first term of 5 and a common ratio of 3. What is the 6th term of the sequence?
["Understanding a Geometric Sequence: Finding the 6th Term with a First Term of 5 and Common Ratio of 3", "When exploring sequences in mathematics, geometric sequences often stand out for their pattern and predictive power. If you’ve been wondering, “What is the 6th term of a geometric sequence with a first term of 5 and a common ratio of 3?” — this article will walk you through the concept and provide a clear, step-by-step solution.", "### What is a Geometric Sequence?", "A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This simple rule creates a powerful pattern that appears in many real-world applications—from finance to science and computer algorithms.", "### The General Formula", "To find any term in a geometric sequence, we use the standard formula:", "[\na_n = a_1 \ imes 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", "### Applying the Values", "Given:\n- First term (a_1 = 5)\n- Common ratio (r = 3)\n- Term number (n = 6)", "Plug these into the formula:", "[\na_6 = 5 \ imes 3^{(6-1)} = 5 \ imes 3^5\n]", "Now calculate (3^5):", "[\n3^5 = 3 \ imes 3 \ imes 3 \ imes 3 \ imes 3 = 243\n]", "Then multiply:", "[\na_6 = 5 \ imes 243 = 1215\n]", "### Conclusion: The 6th Term Is 1,215", "The 6th term of the geometric sequence with first term 5 and common ratio 3 is 1,215. This clear mathematical insight helps build strong comprehension of pattern recognition, exponential growth, and sequence analysis.", "Whether you're solving math problems, studying algebra, or exploring logarithmic trends, understanding geometric sequences and how to compute specific terms is invaluable. Recognizing that each term grows exponentially—multiplied constantly by the ratio—empowers learners to tackle increasingly complex numerical challenges with confidence.", "If you're studying sequences or preparing for standardized math tests, mastering formulas and term calculations is essential. The geometric sequence exemplifies how simple rules generate powerful, predictable patterns. Start now with (a_1 = 5) and (r = 3)—the 6th term awaits!", "---", "Keywords: geometric sequence, 6th term formula, common ratio, exponential growth, mathematical sequence, algebra tutorial, math problem solving, recurring terms, STEM education, sequence calculation.", "Meta Description: Learn how to find the 6th term of a geometric sequence with first term 5 and common ratio 3 using the standard formula. Step-by-step explanation with calculation."]
