The sum is \(rac{a}{1 - r} = rac{5}{1 - rac{1}{3}} = rac{5}{rac{2}{3}} = 5 imes rac{3}{2} = 7.5\).

["# Understanding the Formula: Sum of a Geometric Series — ( S = \dfrac{a}{1 - r} = 7.5 )", "The geometric series represents a powerful mathematical concept widely used in finance, physics, and engineering. One of its key formulas — ( S = \dfrac{a}{1 - r} ) — helps compute the sum of an infinite series where each term decreases by a constant ratio ( r ), provided ( |r| < 1 ). In this article, we’ll explore the step-by-step derivation of this formula using a concrete example and explain its significance.", "## What is a Geometric Series?", "A geometric series is a sequence where each term is obtained by multiplying the previous term by a fixed number called the common ratio, denoted by ( r ). The general term of the series is:\n[ a, , ar, , ar^2, , ar^3, \dots ]\nwhere ( a ) is the first term and ( r ) is the common ratio.", "## When Does the Formula Apply?", "The formula ( S = \dfrac{a}{1 - r} ) applies only when the absolute value of the common ratio is less than one ((|r| < 1)). In this case, the series converges to a finite sum rather than growing indefinitely.", "---", "### Step-by-Step Explanation of the Given Example", "Let’s evaluate the expression:\n[ \dfrac{a}{1 - r} = \dfrac{5}{1 - \dfrac{1}{3}} ]", "1. Substitute values into the formula:\n Given ( a = 5 ) and ( r = \dfrac{1}{3} ), plug these into the sum formula:\n [\n S = \dfrac{5}{1 - \frac{1}{3}}\n ]", "2. Simplify the denominator:\n [\n 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}\n ]", "3. Divide numerator by denominator:\n [\n S = \frac{5}{\frac{2}{3}} = 5 \ imes \frac{3}{2} = \frac{15}{2} = 7.5\n ]", "Thus, the total sum of the infinite geometric series is ( 7.5 ).", "---", "### Why This Formula Matters", "This formula simplifies calculations involving infinite geometric sequences, which appear in:", "- Financial calculations like calculating perpetuities or cumulative investments\n- Physics for modeling wave decay or electric signals\n- Computer science for analyzing algorithm efficiencies", "Without convergence (( |r| < 1 )), the sum would be infinite — making this formula both elegant and practically essential.", "---", "### Final Insight: Condition for Convergence", "Remember:\n- If ( |r| < 1 ), the series converges and the sum is finite: ( S = \dfrac{a}{1 - r} )\n- If ( |r| \geq 1 ), the sum diverges, and the formula does not apply", "---", "In summary, understanding and applying ( S = \dfrac{a}{1 - r} = 7.5 ) demonstrates a foundational skill in mathematical analysis and real-world problem-solving. Whether you're solving for ( a ) or ( r ), or modeling long-term trends, mastering this formula enhances your analytical toolkit.", "---", "Keywords: geometric series formula, sum of geometric series, infinite series formula, mathematical derivation, ( S = \dfrac{a}{1 - r} ), convergence condition, financial math, perpetuity calculation.", "Meta Description:\nLearn how ( \dfrac{a}{1 - r} = \dfrac{5}{1 - \frac{1}{3}} = 7.5 ) illustrates the sum of an infinite geometric series. Understand the formula, its condition (|r| < 1), and real-world applications in finance and physics."]