A geometric sequence has a first term of 5 and a common ratio of 3. What is the sum of the first 6 terms?
["Geometric Sequence: Calculating the Sum of the First 6 Terms with First Term 5 and Common Ratio 3", "A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. One of the most powerful aspects of geometric sequences is the ability to calculate the sum of the first several terms efficiently using a mathematical formula.", "In this article, we’ll explore a specific geometric sequence with:\n- First term ( a = 5 )\n- Common ratio ( r = 3 )\nWe’ll determine the sum of the first 6 terms.", "---", "### What Is a Geometric Sequence?", "A geometric sequence follows this general form:\n[\na, ar, ar^2, ar^3, \ldots\n]\nEach term is obtained by multiplying the earlier term by the common ratio ( r ).", "---", "### Given Values:", "- First term (( a )) = 5\n- Common ratio (( r )) = 3\n- Number of terms (( n )) = 6", "---", "### Formula for the Sum of the First ( n ) Terms", "The sum ( S_n ) of the first ( n ) terms of a geometric sequence is given by:", "[\nS_n = a \frac{r^n - 1}{r - 1} \quad \ ext{(for } r <br/>\ne 1\ ext{)}\n]", "Plugging in our values:", "[\nS_6 = 5 \cdot \frac{3^6 - 1}{3 - 1}\n]", "---", "### Step-by-Step Calculation", "1. Compute ( 3^6 ):\n[\n3^6 = 729\n]", "2. Subtract 1:\n[\n729 - 1 = 728\n]", "3. Divide by ( r - 1 = 3 - 1 = 2 ):\n[\n\frac{728}{2} = 364\n]", "4. Multiply by ( a = 5 ):\n[\nS_6 = 5 \cdot 364 = 1820\n]", "---", "### Conclusion", "The sum of the first 6 terms of the geometric sequence with a first term of 5 and a common ratio of 3 is:", "[\n\boxed{1820}\n]", "This efficient method avoids adding each term individually, making it ideal for sequences with rapidly growing values like this one. Whether used in mathematics, finance, science, or computer science, understanding geometric series and their sums empowers problem-solving across disciplines.", "If you’re working with exponential growth patterns, knowing how to calculate these sums quickly will enhance both your comprehension and practical application. Start with the formula, plug in your values, and solve confidently — your next mathematical challenge just got simpler!"]
