Sum = (rⁿ⁺¹ - 1)/(r - 1) = (2.5⁵ - 1)/(2.5 - 1) = (97.65625 - 1)/1.5 = 96.65625/1.5 ≈ 64.4375
["Understanding the Geometric Series Formula: Sum = (rⁿ⁺¹ - 1)/(r - 1) and Its Practical Calculation", "The formula for the sum of a geometric series is a fundamental concept in mathematics with wide applications in finance, science, and engineering. One of the most widely used forms is:", "[\nS = \frac{r^{n+1} - 1}{r - 1}, \quad \ ext{where } r <br/>\neq 1\n]", "This expression calculates the sum of the first ( n ) terms of a geometric sequence where each term is multiplied by a common ratio ( r > 0 ).", "---", "### What is a Geometric Series?", "A geometric series occurs when each term after the first is found by multiplying the previous term by a constant ratio ( r ). For example, if the first term is ( a ) and the ratio is ( r ), the series is:", "[\na + ar + ar^2 + ar^3 + \cdots + ar^{n-1}\n]", "The sum of the first ( n ) terms is:", "[\nS_n = a \cdot \frac{r^n - 1}{r - 1}\n]", "For simplicity, if the first term ( a = 1 ), the sum becomes:", "[\nS = \frac{r^{n+1} - 1}{r - 1}\n]", "This is the formula we’ll explore in detail with a concrete example.", "---", "### Real-W-life Example: Computing Sum with r = 2.5", "Let’s compute the sum using the formula with specific values:\n( r = 2.5 ), ( n = 5 )", "Using the formula:", "[\nS = \frac{r^{n+1} - 1}{r - 1} = \frac{2.5^{6} - 1}{2.5 - 1} = \frac{97.65625 - 1}{1.5} = \frac{96.65625}{1.5}\n]", "Now divide:", "[\n\frac{96.65625}{1.5} = 64.4375\n]", "Thus,\n[\nS = 64.4375\n]", "---", "### Why Is This Formula Important?", "The geometric series sum formula models exponential growth — common in compound interest calculations, population growth, and digital signal processing. By using this formula, you can efficiently compute sums that would otherwise require time-consuming addition of each term.", "---", "### Step-by-Step Breakdown of the Calculation", "1. Identify the parameters:\n - Common ratio: ( r = 2.5 )\n - Number of terms: ( n = 5 )\n - First term assumed as 1: ( a = 1 )", "2. Apply the geometric series sum formula:\n [\n S = \frac{r^{n+1} - 1}{r - 1} = \frac{2.5^{6} - 1}{2.5 - 1}\n ]", "3. Calculate ( r^{n+1} = 2.5^{6} ):\n ( 2.5^6 = 97.65625 )", "4. Subtract 1:\n ( 97.65625 - 1 = 96.65625 )", "5. Divide by ( r - 1 = 1.5 ):\n ( \frac{96.65625}{1.5} \approx 64.4375 )", "---", "### Summary", "The geometric series sum formula provides a powerful and efficient way to compute the sum of exponential sequences:", "[\n\boxed{S = \frac{r^{n+1} - 1}{r - 1} = 64.4375 \ ext{ when } r = 2.5 \ ext{ and } n = 5}\n]", "Whether used in financial modeling, computer science, or algorithmic analysis, understanding and applying this formula enhances analytical problem-solving skills and computational efficiency.", "---", "Keywords: geometric series, sum formula, rⁿ⁺¹ − 1 / (r − 1), exponential growth, math formula, financial math, compound interest, mathematical applications, exponential sum, n terms series", "Meta Description:\nDiscover how to compute the geometric series sum ( S = \frac{r^{n+1} - 1}{r - 1} ) using a practical example with ( r = 2.5 ), ( n = 5 ), resulting in a sum of approximately 64.4375. Learn the formula, its applications, and step-by-step calculations."]
