A geometric series has a first term of 3 and a common ratio of 2. Find the sum of the first 5 terms.
["Geometric Series: Finding the Sum of the First 5 Terms with First Term 3 and Common Ratio 2", "Understanding geometric series is a fundamental concept in mathematics, especially in algebra and calculus. Whether you’re studying for exams or solving real-world problems, knowing how to calculate the sum of the first few terms can be incredibly useful. In this article, we’ll explore a classic geometric series: one with a first term of 3 and a common ratio of 2. We’ll walk through the steps to find the sum of the first 5 terms, offering practical insight and review of essential formulas.", "---", "### 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 constant called the common ratio. Formally, a geometric series with first term ( a ) and common ratio ( r ) takes the form:", "[\na, ar, ar^2, ar^3, \ldots\n]", "The sum of the first ( n ) terms — known as the partial sum — is given by the formula:", "[\nS_n = a \frac{r^n - 1}{r - 1} \quad \ ext{(if } r <br/>\ne 1\ ext{)}\n]", "This formula works when the common ratio ( r ) is not equal to 1. In cases where ( r = 1 ), the series is simply ( n ) copies of ( a ), so ( S_n = n \cdot a ).", "---", "### Apply the Formula to Our Problem", "We are given:", "- First term ( a = 3 )\n- Common ratio ( r = 2 )\n- Number of terms ( n = 5 )", "Since ( r = 2 <br/>\ne 1 ), we use the formula:", "[\nS_5 = 3 \cdot \frac{2^5 - 1}{2 - 1}\n]", "---", "### Step-by-Step Calculation", "1. Compute ( 2^5 ):", "[\n2^5 = 32\n]", "2. Subtract 1:", "[\n32 - 1 = 31\n]", "3. Divide by ( r - 1 = 2 - 1 = 1 ):", "[\n\frac{31}{1} = 31\n]", "4. Multiply by the first term ( a = 3 ):", "[\nS_5 = 3 \cdot 31 = 93\n]", "---", "### Verification: Sum of Terms Manually", "For verification, let’s compute the first 5 terms explicitly:", "- Term 1: ( 3 \cdot 2^0 = 3 )\n- Term 2: ( 3 \cdot 2^1 = 6 )\n- Term 3: ( 3 \cdot 2^2 = 12 )\n- Term 4: ( 3 \cdot 2^3 = 24 )\n- Term 5: ( 3 \cdot 2^4 = 48 )", "Sum:\n[\n3 + 6 + 12 + 24 + 48 = 93\n]", "This matches our formula-based result, confirming accuracy.", "---", "### Why This Matters", "Mastering geometric series like this one builds a strong foundation for higher-level math. Such questions appear frequently in:", "- Algebra classes\n- Financial mathematics (compound interest, for example)\n- Computer algorithms involving exponential growth\n- Physics models involving repeated multiplicative processes", "By understanding how to compute partial sums, you equip yourself with a powerful tool for analyzing patterns where values grow exponentially.", "---", "### Final Thoughts", "Finding the sum of the first 5 terms of a geometric series with ( a = 3 ) and ( r = 2 ) is straightforward using the formula:", "[\nS_5 = 3 \cdot \frac{2^5 - 1}{2 - 1} = 93\n]", "Whether learned in school or applied in real-world problems, spiral series help unlock deeper patterns in numbers — and now you’re equipped to calculate them with confidence!", "---", "Keywords: geometric series sum, sum of geometric series formula, first term 3, common ratio 2, math examples, partial sum of geometric series, algebra practice, exponential growth calculation."]
