The sum of an infinite geometric series is 20, and the first term is 5. Find the common ratio.
["Title: How to Find the Common Ratio of an Infinite Geometric Series (Sum = 20, First Term = 5)", "When studying infinite geometric series, one of the most essential formulas is used to calculate the sum when the series converges. The formula for the sum ( S ) of an infinite geometric series is:", "[\nS = \frac{a}{1 - r}\n]", "where:\n- ( S ) is the total sum of the series\n- ( a ) is the first term\n- ( r ) is the common ratio (( |r| < 1 ) for convergence)", "In this article, we’ll apply this formula to solve a common problem:\nGiven: The sum of the infinite geometric series is 20, and the first term is 5. Find the common ratio ( r ).", "---", "### Step-by-Step Solution", "We are given:\n- Sum ( S = 20 )\n- First term ( a = 5 )", "Using the convergence formula:", "[\n20 = \frac{5}{1 - r}\n]", "Now solve for ( r ):", "1. Multiply both sides by ( 1 - r ):", "[\n20(1 - r) = 5\n]", "2. Distribute the 20:", "[\n20 - 20r = 5\n]", "3. Subtract 20 from both sides:", "[\n-20r = 5 - 20\n]", "[\n-20r = -15\n]", "4. Divide both sides by -20:", "[\nr = \frac{-15}{-20} = \frac{3}{4}\n]", "---", "### Verifying the Solution", "Since ( |r| = \frac{3}{4} = 0.75 < 1 ), the series converges, and the formula is valid.", "Check the result:", "[\nS = \frac{a}{1 - r} = \frac{5}{1 - 0.75} = \frac{5}{0.25} = 20\n]", "✔ Confirmed. The common ratio is ( \frac{3}{4} ).", "---", "### Why the Common Ratio Matters", "The common ratio determines how each term relates to the previous one. For the infinite series to converge and sum to a finite value, ( |r| < 1 ). Knowing ( r ) helps in analyzing convergence, finding partial sums, and predicting behavior in finance, physics, and engineering.", "---", "### Final Answer", "The common ratio of the infinite geometric series where the first term is 5 and the sum is 20 is:", "[\n\boxed{\frac{3}{4}}\n]", "---", "### SEO Keywords & Meta Tags (for content optimization)", "- Keywords: infinite geometric series sum formula, common ratio infinite series, geometric series converges, solve infinite geometric series\n- Meta Description: Learn how to calculate the common ratio of an infinite geometric series using the sum and first term. Step-by-step example with solution: sum = 20, first term = 5.\n- Header Tags:\n H1: Find the Common Ratio of an Infinite Geometric Series (Sum = 20, First Term = 5)\n H2: Formula and Derivation\n H3: How to Calculate ( r ) Given Sum and First Term\n H4: Verification and Practical Meaning\n- Apartment: Beautiful, clear, educational, keyword-rich article ideal for students, math learners, and educators."]
