The sum of the first \( n \) terms of an arithmetic sequence is given by \( S_n = \frac{n}{2}(2a + (n-1)d) \). If the sum of the first 10 terms is 150 and the first term is 5, what is the common difference \( d \)?
["Understanding the Sum Formula of an Arithmetic Sequence: Finding the Common Difference", "The sum of the first ( n ) terms of an arithmetic sequence is a foundational concept in algebra and series analysis. The general formula is:", "[\nS_n = \frac{n}{2} \left(2a + (n - 1)d\right)\n]", "Where:\n- ( S_n ) is the sum of the first ( n ) terms,\n- ( a ) is the first term,\n- ( d ) is the common difference,\n- ( n ) is the number of terms.", "This formula allows us to quickly compute cumulative sums in arithmetic progressions without adding each term individually.", "---", "A typical problem in algebra classes applies this formula with concrete values to find the unknown common difference. Consider the following scenario:", "> If the sum of the first 10 terms (( S_{10} )) is 150 and the first term ( a ) is 5, what is the common difference ( d )?", "Let’s solve this step by step.", "---", "### Step 1: Plug known values into the sum formula", "Using\n[\nS_n = \frac{n}{2} \left(2a + (n - 1)d\right)\n]\nand substituting ( S_{10} = 150 ), ( n = 10 ), and ( a = 5 ):", "[\n150 = \frac{10}{2} \left(2 \cdot 5 + (10 - 1)d\right)\n]", "Simplify inside the parentheses:", "[\n150 = 5 \left(10 + 9d\right)\n]", "---", "### Step 2: Solve for ( d )", "Expand the right side:", "[\n150 = 5(10 + 9d) = 50 + 45d\n]", "Subtract 50 from both sides:", "[\n100 = 45d\n]", "Divide both sides by 45:", "[\nd = \frac{100}{45} = \frac{20}{9}\n]", "---", "### Final Answer:", "The common difference of the arithmetic sequence is\n[\n\boxed{\frac{20}{9}}\n]", "---", "This example illustrates how the standard sum formula provides an efficient way to find unknown parameters in arithmetic sequences. Mastering this formula and practice with real values helps strengthen algebraic reasoning and problem-solving skills, essential for math exams and higher-level studies."]
