The sum of the first 10 terms of an arithmetic sequence is 150. If the first term is 5, what is the common difference?
["Finding the Common Difference in an Arithmetic Sequence: A Step-by-Step Guide", "Arithmetic sequences are a fundamental concept in mathematics, widely used in fields such as finance, engineering, and data analysis. One intriguing problem often encountered is determining the common difference when the sum of the first several terms is known. This article explores a classic scenario: If the sum of the first 10 terms of an arithmetic sequence is 150 and the first term is 5, what is the common difference?", "---", "### What Is an Arithmetic Sequence?", "An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference, typically denoted by ( d ). The sequence starts with an initial term ( a ), and each subsequent term increases (or decreases) by ( d ).", "The formula for the ( n )-th term is:", "[\na_n = a + (n - 1)d\n]", "The sum of the first ( n ) terms, ( S_n ), is given by:", "[\nS_n = \frac{n}{2} \left(2a + (n - 1)d\right)\n]", "---", "### Given Information", "We are told:", "- The sum of the first 10 terms:\n[\nS_{10} = 150\n]", "- The first term:\n[\na = 5\n]", "We are to find the common difference ( d ).", "---", "### Step 1: Use the Sum Formula", "Plug ( n = 10 ), ( S_{10} = 150 ), and ( a = 5 ) into the sum formula:", "[\nS_{10} = \frac{10}{2} \left(2(5) + (10 - 1)d\right)\n]", "Simplify:", "[\n150 = 5 \left(10 + 9d\right)\n]", "Divide both sides by 5:", "[\n30 = 10 + 9d\n]", "Subtract 10 from both sides:", "[\n20 = 9d\n]", "Solve for ( d ):", "[\nd = \frac{20}{9}\n]", "---", "### Conclusion", "The common difference of the arithmetic sequence is ( \frac{20}{9} ). This solution demonstrates how basic algebraic manipulation of the arithmetic series formula enables us to uncover key parameters merely known from the problem setup.", "Understanding such sequences not only strengthens algebraic skills but also equips learners to model real-world phenomena, from predicting growth trends to calculating recurring payments.", "Keywords: arithmetic sequence, common difference, sum of terms, arithmetic series formula, algebra problem solving.\nMeta description: Learn how to find the common difference in an arithmetic sequence when given the first term and the sum of the first 10 terms. Step-by-step solution with formula."]
