The sum of an infinite geometric series is 12, and the first term is 3. Find the common ratio.
["The Sum of an Infinite Geometric Series: How to Find the Common Ratio (Given the Sum = 12 and First Term = 3)", "If you’ve ever encountered the concept of an infinite geometric series, you might know a powerful formula:", "The sum ( S ) of an infinite geometric series is given by:\n[\nS = \frac{a}{1 - r}\n]\nwhere:\n- ( a ) is the first term,\n- ( r ) is the common ratio, and\n- ( |r| < 1 ) for the series to converge.", "---", "### Problem Statement\nWe are told that:\n- The sum of the infinite geometric series is ( 12 ),\n- The first term ( a = 3 ).", "We need to find the common ratio ( r ).", "---", "### Step-by-Step Solution", "Start with the infinite geometric series sum formula:\n[\nS = \frac{a}{1 - r}\n]", "Substitute the known values:\n[\n12 = \frac{3}{1 - r}\n]", "Now, solve for ( r ):", "1. Multiply both sides by ( 1 - r ):\n[\n12(1 - r) = 3\n]", "2. Distribute:\n[\n12 - 12r = 3\n]", "3. Subtract 12 from both sides:\n[\n-12r = 3 - 12\n]\n[\n-12r = -9\n]", "4. Divide both sides by (-12):\n[\nr = \frac{-9}{-12} = \frac{3}{4}\n]", "---", "### Verifying the Solution", "Check that ( |r| < 1 ):\n[\n\left| \frac{3}{4} \right| = 0.75 < 1\n]\nSo the series converges, and the formula is valid.", "Now verify the sum:\n[\nS = \frac{3}{1 - \frac{3}{4}} = \frac{3}{\frac{1}{4}} = 3 \ imes 4 = 12\n]\n✓ Matches given sum.", "---", "### Conclusion\nThe common ratio of the infinite geometric series with first term 3 and total sum 12 is:\n(\boxed{\frac{3}{4}})", "This illustrates how the geometric series formula elegantly links the first term, common ratio, and total sum — especially for convergence cases where ( |r| < 1 ).", "If you're studying series in mathematics, physics, or finance, knowing how to extract ( r ) from ( S ) and ( a ) is essential for modeling continuous growth or decay processes."]
