Find the sum of the first 15 terms of the arithmetic sequence where the first term is 7 and the common difference is 4.
["Finding the Sum of the First 15 Terms in an Arithmetic Sequence", "When studying sequences in mathematics, one common task is calculating the sum of the first few terms of an arithmetic sequence. Whether you're solving textbook problems or preparing for exams, understanding how to apply the arithmetic sequence sum formula efficiently is essential. In this article, we’ll walk through how to find the sum of the first 15 terms of a specific sequence where the first term is 7 and the common difference is 4.", "---", "### What is an Arithmetic Sequence?", "An arithmetic sequence is a series of numbers where each term increases by a constant difference. This difference is called the common difference. Given:", "- First term ( a = 7 )\n- Common difference ( d = 4 )\n- Number of terms ( n = 15 )", "We want to find the sum ( S_n ) of the first 15 terms.", "---", "### Formula for the Sum of an Arithmetic Sequence", "The sum of the first ( n ) terms of an arithmetic sequence is given by:", "[\nS_n = \frac{n}{2} \left( 2a + (n - 1)d \right)\n]", "Alternatively, if you know the first term and the last term, the formula can also be:", "[\nS_n = \frac{n}{2} \ imes (a + l)\n]", "where ( l ) is the last (15th) term. This version can sometimes simplify calculations.", "---", "### Step-by-Step Calculation", "#### Step 1: Find the 15th Term\nFirst, compute the 15th term ( l ) using the formula:", "[\nl = a + (n - 1)d = 7 + (15 - 1) \ imes 4 = 7 + 56 = 63\n]", "#### Step 2: Apply the Sum Formula\nUsing the sum formula ( S_n = \frac{n}{2}(a + l) ):", "[\nS_{15} = \frac{15}{2} \ imes (7 + 63) = \frac{15}{2} \ imes 70 = 15 \ imes 35 = 525\n]", "---", "### Result", "The sum of the first 15 terms of the arithmetic sequence with first term 7 and common difference 4 is 525.", "---", "### Why This Formula Works", "The arithmetic sequence sum formula averages the first and last terms and multiplies by the number of terms. This method efficiently sums a long sequence without adding each term individually.", "---", "### Practical Applications", "Understanding how to sum arithmetic sequences helps in many real-world situations, such as:", "- Calculating total payments over time\n- Estimating cumulative growth in sequences\n- Solving problems in physics and finance involving linear progressions", "---", "### Final Summary", "- Given: ( a = 7 ), ( d = 4 ), ( n = 15 )\n- 15th term ( l = 63 )\n- Sum ( S_{15} = 525 )", "Mastering this approach enables quick, accurate solutions across a wide range of arithmetic and real-life math challenges.", "---", "Keywords: arithmetic sequence sum, find sum of first 15 terms, arithmetic progression formula, calculate arithmetic sum, n-term sequence sum, common difference 4, first term 7\nMeta Description: Learn how to find the sum of the first 15 terms of an arithmetic sequence with first term 7 and common difference 4 using the formula and step-by-step examples."]
