The sum of an arithmetic series is given by \( S_n = \frac{n}{2}(a + l) \), where \( S_n = 210 \), \( n = 7 \), and \( a = 15 \).

["# The Sum of an Arithmetic Series: How ( S_n = \frac{n}{2}(a + l) ) Solves for Unknown Values", "Understanding the sum of an arithmetic series is fundamental in algebra and real-world problem solving. Whether you're calculating total savings, forecasting revenue, or solving textbook problems, the formula\n$$\nS_n = \frac{n}{2}(a + l)\n$$\nis an essential tool. This article explains how this formula works, demonstrates its application with a real example where ( S_n = 210 ), ( n = 7 ), and ( a = 15 ), and shows how to solve for the unknown last term ( l ).", "---", "## What Is an Arithmetic Series?", "An arithmetic series is the sum of terms in an arithmetic sequence—where each term increases (or decreases) by a constant difference. The general form of an arithmetic sequence is:\n$$\na, a + d, a + 2d, a + 3d, \dots\n$$\nwhere ( a ) is the first term, ( d ) is the common difference, and ( n ) is the number of terms.", "The sum ( S_n ) of the first ( n ) terms is determined by these two key parameters:\n- ( n ): number of terms\n- ( a ) and ( l ): the first and last terms, respectively", "---", "## The Standard Sum Formula", "The formula to compute the sum of the first ( n ) terms of an arithmetic series is:\n$$\nS_n = \frac{n}{2}(a + l)\n$$", "This formula works because it averages the first and last term and multiplies by the number of terms—efficiently capturing the cumulative total.", "---", "## Applying the Formula with Real Numbers", "Let’s apply this to a concrete example where:\n- Total sum ( S_n = 210 )\n- Number of terms ( n = 7 )\n- First term ( a = 15 )\n- Unknown: last term ( l )", "Using the sum formula:\n$$\n210 = \frac{7}{2}(15 + l)\n$$", "Now solve for ( l ):\nFirst, multiply both sides by 2 to eliminate the denominator:\n$$\n420 = 7(15 + l)\n$$", "Next, divide both sides by 7:\n$$\n60 = 15 + l\n$$", "Finally, subtract 15 from both sides:\n$$\nl = 45\n$$", "Result: The last term of the series is ( 45 ).", "---", "## Verifying the Series", "To confirm, reconstruct the series using ( a = 15 ), ( l = 45 ), and ( n = 7 ):\nList the terms (since ( d = \frac{l - a}{n - 1} = \frac{30}{6} = 5 )):\n$$\n15, 20, 25, 30, 35, 40, 45\n$$", "Sum them:\n$$\n15 + 20 + 25 + 30 + 35 + 40 + 45 = 210\n$$\nThe total matches perfectly.", "---", "## Why This Formula Matters", "This formula simplifies solving for missing values in arithmetic sequences:\n- Use it when total sum and number of terms are known, but last term is unknown.\n- It supports educational learning, financial planning (e.g., total interest over time), and data analysis.", "---", "## Conclusion", "The sum formula ( S_n = \frac{n}{2}(a + l) ) is a powerful shortcut in arithmetic and beyond. Knowing how to apply it—whether you’re a student or a professional—enhances problem-solving speed and accuracy. In this example, from ( S_n = 210 ), ( n = 7 ), and ( a = 15 ), we found ( l = 45 ), validating both algebra and arithmetic logic.", "Practice Tip: Try using this formula with different values—plot the series mentally or write out terms to deepen understanding. Mastering it opens doors to more advanced math and real-world computation.", "---", "### Key Takeaways:\n- Formula: ( S_n = \frac{n}{2}(a + l) )\n- Use when: Sum, number of terms, and first term are known; last term is unknown.\n- Example solved: ( S_7 = 210 ), ( a = 15 ) → ( l = 45 )\n- Verify: Reconstruct series and confirm total matches", "By mastering this formula, you build a strong foundation for algebra, finance, and data-driven decision-making.", "---", "Related Keywords: arithmetic series formula, sum of arithmetic sequence, solve arithmetic series, ( S_n = \frac{n}{2}(a + l) ), algebra tips, math problem solving, series total, common difference, first and last term.", "---", "Meta Description for SEO:\nLearn how to calculate the sum of an arithmetic series using ( S_n = \frac{n}{2}(a + l) ) with real numbers. This guide solves ( S_n = 210 ), ( n = 7 ), ( a = 15 ), showing ( l = 45 ) and verifying the result step-by-step. Ideal for algebra students and practical problem-solving."]