The sum of the first 10 terms of an arithmetic sequence is 155. If the first term is 5, find the common difference.
["Title: How to Find the Common Difference in an Arithmetic Sequence — Given the Sum of the First 10 Terms", "---", "Meta Description:\nUnlock how to calculate the common difference in an arithmetic sequence when you know the first term and the sum of the first 10 terms. Solve this classic problem step-by-step using the arithmetic series formula.", "---", "## Introduction", "Arithmetic sequences are fundamental in mathematics, appearing in everything from finance to physics. One common problem involves finding the common difference ( d ) when provided the first term and the sum of the first several terms. In this article, we’ll explore how to determine the common difference when the first term is 5 and the sum of the first 10 terms equals 155.", "---", "## Understanding Arithmetic Sequences", "An arithmetic sequence is a set of numbers where each term increases by a constant value—the common difference ( d ). The sequence begins with a first term ( a ), and the next terms are:\n( a, a+d, a+2d, a+3d, \dots )", "The ( n^{\ ext{th}} ) term is given by:\n[\na_n = a + (n-1)d\n]", "The sum of the first ( n ) terms of an arithmetic sequence is:\n[\nS_n = \frac{n}{2} \left(2a + (n-1)d\right)\n]", "---", "## Problem Setup", "We are told:\n- First term ( a = 5 )\n- Sum of the first 10 terms ( S_{10} = 155 )\n- Find the common difference ( d )", "Using the sum formula:\n[\nS_n = \frac{n}{2} \left(2a + (n-1)d\right)\n]", "Substitute ( n = 10 ), ( a = 5 ), and ( S_{10} = 155 ):\n[\n155 = \frac{10}{2} \left(2 \cdot 5 + (10 - 1)d \right)\n]\n[\n155 = 5 \left(10 + 9d \right)\n]\n[\n155 = 50 + 45d\n]", "Subtract 50 from both sides:\n[\n105 = 45d\n]", "Divide both sides by 45:\n[\nd = \frac{105}{45} = \frac{7}{3}\n]", "---", "## Final Answer", "The common difference of the arithmetic sequence is ( \boxed{\frac{7}{3}} ).", "---", "## Why This Matters", "Knowing how to calculate the common difference strengthens your ability to work with sequences in real-world applications—such as modeling steady growth, analyzing periodic data, or solving algebraic proofs. Mastering the arithmetic series formula is a critical step in advancing in algebra and discrete mathematics.", "---", "Keywords: arithmetic sequence sum, find common difference, arithmetic progression formula, sum of first 10 terms, solve arithmetic sequence, math problem solving", "Tags: #ArithmeticSequence #MathTips #Algebra #CommonDifference #SumOfTerms #Education", "---", "### Summary\nBy applying the arithmetic series formula with the known first term and sum, we easily found that the common difference is ( \frac{7}{3} ). This practical example shows how foundational concepts yield powerful results.", "---", "If you’re learning or teaching sequences, practice similar problems to build confidence—every sum leads you closer to true mastery."]
