The sum of an arithmetic series is 300, with 10 terms and the first term is 10. What is the common difference?
["Finding the Common Difference in an Arithmetic Series: Sum = 300, First Term = 10, Number of Terms = 10", "If you've ever studied arithmetic sequences, you’ve likely encountered the formula for the sum of the first n terms. One classic problem involves finding the common difference when the sum, number of terms, and first term are given — a perfect exercise to sharpen your understanding.", "In this article, we’ll solve a practical problem:\nThe sum of an arithmetic series is 300, with 10 terms and the first term being 10. What is the common difference?", "---", "### Recap: The Sum Formula for an Arithmetic Series", "The sum $ S_n $ of the first $ n $ terms of an arithmetic sequence is given by:", "[\nS_n = \frac{n}{2} \left( 2a + (n-1)d \right)\n]", "where:\n- $ S_n $ = sum of the first $ n $ terms\n- $ n $ = number of terms\n- $ a $ = first term\n- $ d $ = common difference", "---", "### Plugging in the Known Values", "From the problem:\n- $ S_{10} = 300 $\n- $ n = 10 $\n- $ a = 10 $", "Substitute these into the sum formula:", "[\n300 = \frac{10}{2} \left( 2(10) + (10-1)d \right)\n]", "Simplify step-by-step:", "[\n300 = 5 \left( 20 + 9d \right)\n]", "[\n300 = 100 + 45d\n]", "Subtract 100 from both sides:", "[\n200 = 45d\n]", "Solve for $ d $:", "[\nd = \frac{200}{45} = \frac{40}{9}\n]", "---", "### Conclusion", "The common difference of the arithmetic series is $ \frac{40}{9} $. This result demonstrates how algebraic reasoning and the sum formula work together to unlock unknown parameters in sequences.", "---", "### Bonus: Double-Check with the Second Formula", "To confirm, we can use the alternative form:", "[\nS_n = \frac{n}{2} \left( a + l \right)\n]", "where $ l $ is the last term. Since $ a = 10 $, $ n = 10 $, and $ S_n = 300 $, find the last term $ l $:", "[\n300 = \frac{10}{2} (10 + l) \Rightarrow 300 = 5(10 + l) \Rightarrow 60 = 10 + l \Rightarrow l = 50\n]", "Now, recall that in arithmetic sequences:", "[\nl = a + (n-1)d \Rightarrow 50 = 10 + 9d \Rightarrow 9d = 40 \Rightarrow d = \frac{40}{9}\n]", "Consistent results confirm our solution is accurate.", "---", "Key Takeaways:\n- Known the sum, number of terms, and first term.\n- Applied the sum formula $ S_n = \frac{n}{2}(2a + (n-1)d) $.\n- Solved algebraically for $ d $.\n- Verified using the last term method.", "Understanding how to find the common difference in an arithmetic series boosts your math skills and prepares you for advanced sequence problems in algebra and beyond!"]
