A geometric series has a first term of 5 and a common ratio of 2. What is the sum of the first 6 terms?
["# Geometric Series: Finding the Sum of the First 6 Terms", "Understanding geometric series is essential in mathematics, especially in fields like finance, physics, and engineering. A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. In this article, we’ll explore how to calculate the sum of the first six terms of a geometric series with a first term of 5 and a common ratio of 2.", "## What is a Geometric Series?", "A geometric series consists of terms like:\n[\na, ar, ar^2, ar^3, \ldots, ar^{n-1}\n]\nwhere:\n- ( a ) is the first term\n- ( r ) is the common ratio\n- ( n ) is the number of terms", "The formula to calculate the sum ( S_n ) of the first ( n ) terms is:", "[\nS_n = a \cdot \frac{r^n - 1}{r - 1} \quad \ ext{(when } r <br/>\ne 1\ ext{)}\n]", "This formula is derived from the pattern of successive multiplications and allows us to determine the total sum efficiently without adding each term individually.", "## Given Values", "For our specific problem:", "- First term ( a = 5 )\n- Common ratio ( r = 2 )\n- Number of terms ( n = 6 )", "## Step-by-Step Calculation", "Using the geometric series sum formula:", "[\nS_6 = 5 \cdot \frac{2^6 - 1}{2 - 1}\n]", "First, calculate ( 2^6 ):", "[\n2^6 = 64\n]", "Now substitute:", "[\nS_6 = 5 \cdot \frac{64 - 1}{1} = 5 \cdot 63 = 315\n]", "## Conclusion", "The sum of the first 6 terms of a geometric series with a first term of 5 and a common ratio of 2 is:", "[\n\boxed{315}\n]", "This straightforward calculation demonstrates how powerful the geometric series formula can be in solving real-world problems involving exponential growth—such as compound interest or population modeling—giving you a powerful tool to master in mathematics and its applications."]
