Find the sum of the infinite geometric series with first term 5 and common ratio \( rac{1}{3}\).
["# Find the Sum of the Infinite Geometric Series with First Term 5 and Common Ratio ( \frac{1}{3} )", "When studying infinite series, one of the most elegant and frequently encountered results is the sum of an infinite geometric series. In this article, we’ll explore how to find the sum of an infinite geometric series with a specific first term and common ratio, using the example of a first term of 5 and a common ratio of ( \frac{1}{3} ).", "## What Is an Infinite Geometric Series?", "An infinite geometric series is a series of the form:\n[\nS = a + ar + ar^2 + ar^3 + \cdots\n]\nwhere:\n- ( a ) is the first term,\n- ( r ) is the common ratio (( |r| < 1 ) for convergence).", "If the absolute value of the common ratio is less than 1, the series converges, and its sum can be calculated using the formula:\n[\nS = \frac{a}{1 - r}\n]", "## Given Values", "For our problem:\n- First term ( a = 5 )\n- Common ratio ( r = \frac{1}{3} ), which satisfies ( |r| < 1 ), ensuring the series converges.", "## Applying the Infinite Geometric Series Formula", "Substitute ( a = 5 ) and ( r = \frac{1}{3} ) into the sum formula:", "[\nS = \frac{a}{1 - r} = \frac{5}{1 - \frac{1}{3}}\n]", "Simplify the denominator:\n[\n1 - \frac{1}{3} = \frac{2}{3}\n]", "So the sum becomes:\n[\nS = \frac{5}{\frac{2}{3}} = 5 \ imes \frac{3}{2} = \frac{15}{2} = 7.5\n]", "## Conclusion", "The sum of the infinite geometric series with first term 5 and common ratio ( \frac{1}{3} ) is:", "[\n\boxed{\frac{15}{2}} \quad \ ext{or} \quad \boxed{7.5}\n]", "This result demonstrates how an infinite sequence of decreasing terms can converge gracefully to a finite limit—perfect for modeling scenarios like discounted cash flows or decay processes.", "## SEO Keywords:\ninfinite geometric series sum, sum of infinite geometric series, first term 5, common ratio 1/3, formula derivation, convergence of geometric series, mathematical series solution", "---", "This SEO-optimized article clearly explains the concept, applies the formula step-by-step, and presents a clean final result — making it valuable for students, educators, and anyone learning infinite series."]
