The sum of the first 10 terms of an arithmetic sequence is 145. If the first term is 5, what is the common difference?
["Understanding Arithmetic Sequences: Finding the Common Difference", "Arithmetic sequences are one of the most fundamental concepts in mathematics, widely used in algebra, finance, science, and everyday problem solving. One common question students encounter is how to determine the common difference of an arithmetic sequence when given the sum of a certain number of terms and the first term. In this article, we’ll explore how to calculate the common difference when the sum of the first 10 terms is 145, starting from the first term being 5.", "### What Is an Arithmetic Sequence?", "An arithmetic sequence is a sequence of numbers where each term after the first is obtained by adding a constant value called the common difference, denoted as ( d ). The structure of such a sequence is defined by:", "- First term: ( a )\n- Common difference: ( d )\n- General term: ( a_n = a + (n - 1)d )\n- Sum of the first ( n ) terms: ( S_n = \frac{n}{2} \left(2a + (n - 1)d \right) )", "### Given Information", "We are told:\n- The first term ( a = 5 )\n- The sum of the first 10 terms ( S_{10} = 145 )\n- We need to find the common difference ( d )", "### Using the Sum Formula", "The formula for the sum of the first ( n ) terms of an arithmetic sequence is:", "[\nS_n = \frac{n}{2} \left(2a + (n - 1)d \right)\n]", "Plugging in ( n = 10 ), ( a = 5 ), and ( S_{10} = 145 ):", "[\n145 = \frac{10}{2} \left(2(5) + (10 - 1)d \right)\n]", "Simplify:", "[\n145 = 5 \left(10 + 9d \right)\n]", "[\n145 = 50 + 45d\n]", "Subtract 50 from both sides:", "[\n95 = 45d\n]", "Now, solve for ( d ):", "[\nd = \frac{95}{45} = \frac{19}{9}\n]", "### Final Answer", "The common difference of the arithmetic sequence is ( \frac{19}{9} ), or approximately 2.11. This value ensures that the sum of the first 10 terms equals 145 when starting from 5.", "### Why This Matters", "Understanding how to find the common difference using the sum formula strengthens problem-solving skills applicable in various real-life scenarios—from calculating interest growth in annuities to predicting sequential events in sports statistics. Mastering arithmetic sequences lays a strong foundation in algebra that enables deeper exploration into series, functions, and mathematical modeling.", "---", "If you’re learning arithmetic sequences or trying to solve similar word problems, remember to always use the correct sum formula and carefully substitute known values to isolate the unknown common difference. This approach simplifies solving and ensures accurate results."]
