A geometric series has first term 5 and common ratio 3. Find the sum of the first 4 terms.
["Geometric Series: Find the Sum of the First 4 Terms with First Term 5 and Common Ratio 3", "In mathematics, geometric series are powerful tools used in various fields including finance, physics, and engineering. Understanding how to calculate the sum of a geometric series is essential for solving problems involving repeated growth or decay. In this article, we’ll explore a key application: finding the sum of the first four terms of a geometric series where the first term is 5 and the common ratio is 3.", "---", "### What Is a Geometric Series?", "A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form is:", "[\na, ar, ar^2, ar^3, \ldots\n]", "where:\n- ( a ) = first term (also called the initial term),\n- ( r ) = common ratio,\n- ( n ) = number of terms.", "The sum of the first ( n ) terms of a geometric series is given by the formula:", "[\nS_n = a \frac{r^n - 1}{r - 1} \quad \ ext{(when } r <br/>\ne 1 \ ext{)}\n]", "---", "### Given Problem", "We are given:", "- First term ( a = 5 )\n- Common ratio ( r = 3 )\n- Number of terms ( n = 4 )", "We want to find the sum of the first 4 terms:\n[\nS_4 = a + ar + ar^2 + ar^3\n]", "---", "### Step-by-Step Calculation", "#### Step 1: List the first 4 terms\nUsing the formula ( ar^{k} ), compute each term:", "- First term: ( ar^0 = 5 \ imes 3^0 = 5 \ imes 1 = 5 )\n- Second term: ( ar^1 = 5 \ imes 3 = 15 )\n- Third term: ( ar^2 = 5 \ imes 3^2 = 5 \ imes 9 = 45 )\n- Fourth term: ( ar^3 = 5 \ imes 3^3 = 5 \ imes 27 = 135 )", "#### Step 2: Add the terms directly for verification", "[\nS_4 = 5 + 15 + 45 + 135 = 200\n]", "#### Step 3: Use the geometric series sum formula for accuracy", "[\nS_4 = 5 \cdot \frac{3^4 - 1}{3 - 1} = 5 \cdot \frac{81 - 1}{2} = 5 \cdot \frac{80}{2} = 5 \cdot 40 = 200\n]", "Both methods confirm the sum.", "---", "### Why This Matters", "Geometric series with integer ratios like 3 are commonly used in modeling exponential growth—such as population growth, interest compounding, and certain computational algorithms. Learning to compute sums efficiently helps solve real-world problems with compound behavior.", "---", "### Final Answer", "The sum of the first 4 terms of the geometric series with first term 5 and common ratio 3 is:", "[\n\boxed{200}\n]", "---", "Boost Your Math Skills\nUnderstanding geometric series unlocks deeper insights into pattern recognition and exponential change. Practice computing sums like this to strengthen your analytical skills—essential for exams, finance, and science!"]
