A geometric series has a first term of 5 and a common ratio of 2. Find the sum of the first 4 terms.
["Title: How to Calculate the Sum of the First 4 Terms of a Geometric Series with First Term 5 and Common Ratio 2", "### Introduction\nUnderstanding geometric series is essential in mathematics, finance, and engineering. A geometric series is a sum of terms where each term is obtained by multiplying the previous term by a constant called the common ratio. In this article, we’ll explore a specific geometric series: one with a first term of 5 and a common ratio of 2. We’ll walk through the step-by-step calculation to find the sum of the first 4 terms, showing how this simple concept applies to real-world problems.", "### What is a Geometric Series?\nA geometric series is defined by:\n- A first term: ( a )\n- A common ratio: ( r ) (the factor by which each term increases)", "The terms in the series are:\n[ a, ar, ar^2, ar^3, \ldots ]\nThe sum of the first ( n ) terms is given by the formula:\n[ S_n = a \frac{r^n - 1}{r - 1} ]\n(This formula applies when ( r <br/>\ne 1 ).)", "In our example:\n- First term ( a = 5 )\n- Common ratio ( r = 2 )\n- Number of terms ( n = 4 )", "### Step-by-Step: Calculating the Sum of the First 4 Terms", "#### Step 1: List the first 4 terms manually\nUsing ( a = 5 ) and ( r = 2 ), compute each term:\n- Term 1: ( ar^0 = 5 \ imes 2^0 = 5 \ imes 1 = 5 )\n- Term 2: ( ar^1 = 5 \ imes 2^1 = 5 \ imes 2 = 10 )\n- Term 3: ( ar^2 = 5 \ imes 2^2 = 5 \ imes 4 = 20 )\n- Term 4: ( ar^3 = 5 \ imes 2^3 = 5 \ imes 8 = 40 )", "So the first four terms are: 5, 10, 20, 40", "#### Step 2: Add the terms directly\n[ S_4 = 5 + 10 + 20 + 40 = 75 ]", "#### Step 3: Use the geometric series sum formula\n[ S_n = a \frac{r^n - 1}{r - 1} ]\nPlug in the values:\n[ S_4 = 5 \cdot \frac{2^4 - 1}{2 - 1} = 5 \cdot \frac{16 - 1}{1} = 5 \cdot 15 = 75 ]", "### Result\nThe sum of the first 4 terms of the geometric series with first term 5 and common ratio 2 is 75.", "### Why This Matters\nGeometric series appear in contexts like compound interest calculations, population growth models, and fractals. Knowing how to compute sums efficiently helps solve problems involving exponential growth or decay in a straightforward way.", "### Final Notes\n- When the common ratio ( r > 1 ), the terms grow rapidly (as in this example).\n- The formula simplifies calculations and avoids manual addition errors.\n- This method applies to any geometric sequence, making it a powerful tool in math and applied sciences.", "Mastering geometric series not only improves mathematical fluency but also builds a foundation for advanced topics in calculus, finance, and data science. Start with basic examples like this one—your next problem will be just as clear!", "---\nKeywords: geometric series, sum of geometric series, first term 5, common ratio 2, geometric series formula, sum of first 4 terms, exponential growth, math tutorial, exponential series, compound interest, mathematical formulas."]
