Geometric series: a = 12, r = 1.08, n = 6
["# Geometric Series: Understanding the Formula with a = 12, r = 1.08, n = 6", "A geometric series is a powerful concept in mathematics that appears in finance, science, and everyday calculations. Whether you're modeling compound interest or projecting growth, understanding the geometric series formula helps solve real-world problems efficiently. In this article, we’ll explore the geometric series using specific values: ( a = 12 ), ( r = 1.08 ), and ( n = 6 ), and explain how to calculate the sum and individual terms.", "## What is a Geometric Series?", "A geometric series is the sum of the terms in a geometric sequence, where each term is obtained by multiplying the previous term by a constant ratio ( r ). The general form of the geometric series is:", "[\nS_n = a + ar + ar^2 + ar^3 + \dots + ar^{n-1}\n]", "Here:\n- ( a ) = first term\n- ( r ) = common ratio\n- ( n ) = number of terms", "The sum of the first ( n ) terms of a geometric series is given by:", "[\nS_n = a \frac{r^n - 1}{r - 1} \quad \ ext{(when } r <br/>\ne 1\ ext{)}\n]", "---", "## Plugging in the Values: a = 12, r = 1.08, n = 6", "Given:\n- First term: ( a = 12 )\n- Common ratio: ( r = 1.08 )\n- Number of terms: ( n = 6 )", "### Step 1: Compute the 6th Term (ar^{n−1})", "[\na_6 = ar^{5} = 12 \ imes (1.08)^5\n]", "Calculate ( (1.08)^5 ):", "[\n(1.08)^5 \approx 1.46933\n]", "[\na_6 \approx 12 \ imes 1.46933 = 17.632\n]", "The 6th term is approximately 17.632.", "### Step 2: Calculate the Sum of the First 6 Terms, ( S_6 )", "Using the sum formula:", "[\nS_6 = a \frac{r^6 - 1}{r - 1}\n]", "Calculate ( r^6 = (1.08)^6 ):", "[\n(1.08)^6 \approx 1.58687\n]", "Now plug into the formula:", "[\nS_6 = 12 \ imes \frac{1.58687 - 1}{1.08 - 1} = 12 \ imes \frac{0.58687}{0.08}\n]", "[\nS_6 = 12 \ imes 7.3359 \approx 88.031\n]", "Thus, the total sum of the geometric series over 6 terms is approximately 88.031.", "---", "## Why Use a Geometric Series in Real Life?", "One common application is calculating compound interest. When money grows by a fixed percentage each period, the value at each period forms a geometric sequence. For example, investing $12 at an 8% annual growth rate, compounded yearly, results in the sum of these growing amounts over 6 years, which is precisely what ( S_6 \approx 88.03 ) represents.", "---", "## Summary", "| Parameter | Value |\n|-----------|--------|\n| First term ((a)) | 12 |\n| Common ratio ((r)) | 1.08 |\n| Number of terms ((n)) | 6 |\n| 6th term (( ar^5 )) | ~17.632 |\n| Sum of first 6 terms (( S_6 )) | ~88.031 |", "Using the geometric series sum formula:\n[\nS_n = a \frac{r^n - 1}{r - 1}\n]", "enables quick and accurate computations for sequential growth scenarios.", "---", "## Final Thoughts", "Whether you’re learning algebra, analyzing investments, or solving applied math problems, understanding geometric series and how to apply it with your values—for example ( a = 12 ), ( r = 1.08 ), ( n = 6 )—is essential. With practice, calculating series sums becomes intuitive and immensely useful in both academic and real-world settings.", "---", "Keywords: geometric series, geometric progression, series sum formula, compound interest calculation, ( S_n = a \frac{r^n - 1}{r - 1} ), a = 12, r = 1.08, n = 6, mathematical series, exponential growth."]
