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 first term \( a = 5 \) and the common difference \( d = 3 \), find the sum of the first 15 terms.
["Learn How to Calculate the Sum of the First 15 Terms in an Arithmetic Sequence", "Understanding the sum of the first ( n ) terms of an arithmetic sequence is a fundamental concept in algebra and mathematics. The formula for this sum 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, and\n- ( n ) is the number of terms.", "### Applying the Formula to Real-World Problems", "This formula is widely used in financial planning, computer algorithms, and data analysis. For example, it helps calculate total payments in installment plans or cumulative growth in linear trends.", "---", "### Problem Example: Find the Sum of the First 15 Terms", "Let’s apply the formula with specific values:\n- First term ( a = 5 )\n- Common difference ( d = 3 )\n- Number of terms ( n = 15 )", "Step 1: Plug values into the sum formula", "[\nS_{15} = \frac{15}{2} \left(2(5) + (15 - 1)(3)\right)\n]", "Step 2: Simplify inside the parentheses", "[\nS_{15} = \frac{15}{2} \left(10 + 14 \ imes 3\right) = \frac{15}{2} \left(10 + 42\right) = \frac{15}{2} \ imes 52\n]", "Step 3: Complete the calculation", "[\nS_{15} = \frac{15 \ imes 52}{2} = \frac{780}{2} = 390\n]", "---", "### Final Answer", "The sum of the first 15 terms of this arithmetic sequence is 390.", "Using the formula efficiently saves time and reduces errors, making it an essential tool for solving sequence-related problems quickly and accurately.", "---", "Key Takeaways:\n- Use the sum formula ( S_n = \frac{n}{2}(2a + (n-1)d) ) for arithmetic sequences.\n- Identify ( a ), ( d ), and ( n ) clearly before plugging in numbers.\n- Practice with different values to build fluency.", "Start mastering arithmetic sequences today and boost your problem-solving skills!"]
