The sum of an infinite geometric series is 5, and the first term is 2. Find the common ratio.
["The Sum of an Infinite Geometric Series: How to Find the Common Ratio (Given the Sum and First Term)", "When tackling one of the classic problems in algebra, understanding the sum of an infinite geometric series is essential. One common question in math courses is: The sum of an infinite geometric series is 5, and the first term is 2. What is the common ratio? In this SEO-optimized article, we’ll break down the formula, solve for the common ratio, and explain its real-life relevance.", "---", "### What Is the Sum of an Infinite Geometric Series?", "An infinite geometric series occurs when a number is repeatedly multiplied by a constant factor, called the common ratio (denoted as r), resulting in terms that approach zero as the series continues indefinitely. The infinite geometric series formula is:", "$$\nS = \frac{a}{1 - r}\n$$", "where:\n- S is the sum of the series\n- a is the first term\n- r is the common ratio, where |r| < 1 for convergence", "---", "### Given Values in the Problem", "- Sum $ S = 5 $\n- First term $ a = 2 $\n- Find: common ratio $ r $", "---", "### Applying the Formula", "Substitute the known values into the sum formula:", "$$\n5 = \frac{2}{1 - r}\n$$", "Multiply both sides by $ 1 - r $ to eliminate the denominator:", "$$\n5(1 - r) = 2\n$$", "Distribute:", "$$\n5 - 5r = 2\n$$", "Subtract 5 from both sides:", "$$\n-5r = 2 - 5 = -3\n$$", "Divide by -5:", "$$\nr = \frac{-3}{-5} = \frac{3}{5}\n$$", "---", "### Verifying the Common Ratio", "Check that |r| < 1 for convergence:\n$ \left| \frac{3}{5} \right| = 0.6 < 1 $ — this confirms the formula applies and the series converges.", "Now, plug r back into the sum formula to verify:", "$$\nS = \frac{2}{1 - \frac{3}{5}} = \frac{2}{\frac{2}{5}} = 2 \ imes \frac{5}{2} = 5\n$$", "✔️ Correct — the sum matches the given value.", "---", "### Why This Matters in Real Life", "Understanding infinite geometric series helps in finance (e.g., calculating perpetuities), physics (wave damping models), and computer graphics. Knowing how to isolate the common ratio is a vital algebra skill for students and professionals.", "---", "### Final Answer", "The common ratio of the infinite geometric series with sum 5 and first term 2 is:", "$$\n\boxed{\frac{3}{5}}\n$$", "---", "Keywords for SEO Optimization:\ninfinite geometric series sum formula, find common ratio geometric series, sum of infinite geometric series with known first term, algebra problem solution, common ratio algebra, infinite series convergence", "Meta Description:\nLearn how to find the common ratio of an infinite geometric series when the sum and first term are known. Step-by-step solution for sum = 5 and first term = 2 using the formula $ S = \frac{a}{1 - r} $. Join math students and professionals in mastering convergence and series sums.", "---", "Start applying these principles today and make infinite series simpler!"]
