The sum of an arithmetic series with 10 terms is 150, and the first term is 5. Find the common difference.
["Title: How to Find the Common Difference in an Arithmetic Series: A Step-by-Step Guide Using Known Sum and First Term", "---", "Understanding the Sum of an Arithmetic Series is a foundational concept in algebra that has practical applications in finance, programming, and data analysis. In this article, we’ll solve a common problem: Given the sum of the first 10 terms of an arithmetic series is 150, and the first term is 5, determine the common difference.", "Whether you’re a student, educator, or self-learner, mastering this problem helps deepen your understanding of arithmetic sequences.", "---", "### What Is an Arithmetic Series?", "An arithmetic series is the sum of terms in an arithmetic sequence. Each term increases (or decreases) by a constant value called the common difference, denoted here as ( d ).", "The general form of the ( n^{\ ext{th}} ) term is:\n[\na_n = a_1 + (n - 1)d\n]\nAnd the sum of the first ( n ) terms is:\n[\nS_n = \frac{n}{2} \left(2a_1 + (n - 1)d\right)\n]", "---", "### Given Information", "- Number of terms: ( n = 10 )\n- Sum of the series: ( S_{10} = 150 )\n- First term: ( a_1 = 5 )\n- Common difference: ( d = ? )", "---", "### Applying the Sum Formula", "Plug the known values into the sum formula:\n[\nS_n = \frac{n}{2} \left(2a_1 + (n - 1)d\right)\n]", "Substitute ( S_{10} = 150 ), ( n = 10 ), and ( a_1 = 5 ):\n[\n150 = \frac{10}{2} \left(2 \cdot 5 + (10 - 1)d\right)\n]\n[\n150 = 5 \left(10 + 9d\right)\n]", "---", "### Solve for ( d )", "First, divide both sides by 5:\n[\n\frac{150}{5} = 10 + 9d\n]\n[\n30 = 10 + 9d\n]", "Subtract 10 from both sides:\n[\n20 = 9d\n]", "Now divide by 9:\n[\nd = \frac{20}{9}\n]", "---", "### Conclusion", "The common difference in the arithmetic series is ( \boxed{\frac{20}{9}} ).", "This value ensures that each successive term increases by ( \frac{20}{9} ), producing a sequence whose sum of the first 10 terms equals 150, starting from 5.", "---", "### Why This Matters", "Knowing how to calculate the common difference helps solve real-world problems such as calculating recurring payments, estimating cumulative growth, or modeling linear trends with precision. Practicing problems like this strengthens algebraic reasoning and prepares learners for advanced math and programming challenges.", "---", "Keywords: arithmetic series, common difference, sum of arithmetic series, arithmetic sequence formula, solve for d, algebra problem, math tutorial, linear sequences.", "---", "Need more practice? Try calculating the last term or total sum for different values of ( d ) — it’s a great way to deepen understanding!"]
