The sum of an infinite geometric series is 16, and the first term is 8. What is the common ratio?
["The Sum of an Infinite Geometric Series: Finding the Common Ratio When Summing to 16 with a First Term of 8", "Understanding infinite geometric series is essential in mathematics and has wide applications in finance, physics, and engineering. One key formula in this study is the sum formula for an infinite geometric series:", "[\nS = \frac{a}{1 - r}\n]", "where:\n- ( S ) is the sum of the series,\n- ( a ) is the first term,\n- ( r ) is the common ratio, with ( |r| < 1 ).", "In this article, we explore a classic problem:\nThe sum of an infinite geometric series is 16, and the first term is 8. What is the common ratio?", "---", "### The Formula Recap", "Given:\n[\nS = 16, \quad a = 8\n]", "Plug into the infinite sum formula:", "[\n16 = \frac{8}{1 - r}\n]", "### Solving for ( r )", "Multiply both sides by ( 1 - r ):", "[\n16(1 - r) = 8\n]", "Divide both sides by 16:", "[\n1 - r = \frac{8}{16} = 0.5\n]", "Now solve for ( r ):", "[\nr = 1 - 0.5 = 0.5\n]", "---", "### Verification", "Check:\n[\nS = \frac{8}{1 - 0.5} = \frac{8}{0.5} = 16\n]", "This confirms the result is correct.", "---", "### Key Insight: The Validity of the Common Ratio", "For an infinite geometric series to converge and yield a finite sum, the absolute value of the common ratio must be less than 1:\n[\n|r| < 1\n]", "Here, ( r = 0.5 ), so the condition is satisfied — the series converges, and the sum is indeed 16.", "---", "### Conclusion", "In the infinite geometric series where the first term is 8 and the sum converges to 16, the common ratio is 0.5. This problem beautifully illustrates how a simple algebraic manipulation leads directly to the solution, highlighting the elegance and utility of geometric series in mathematics.", "Whether you're solving classroom problems or applying theory in real-world modeling, knowing how to find the common ratio from the sum and first term is an invaluable skill.", "---", "Keywords: infinite geometric series sum formula, common ratio calculation, cooperate ratio from sum and first term, solving geometric series equation, converge infinite series, math problem solving, convergence condition ( |r| < 1 )", "Meta description:\nLearn how to find the common ratio of an infinite geometric series with sum 16 and first term 8 using the formula ( S = \frac{a}{1 - r} ). Step-by-step solution and verification included."]
