Find the sum of the first 8 terms of the arithmetic sequence with first term 5 and common difference 3.
["Find the Sum of the First 8 Terms of an Arithmetic Sequence", "When learning arithmetic sequences, one of the most common and essential tasks is calculating the sum of the first several terms. Understanding this concept helps students and math enthusiasts solve real-world problems efficiently. In this article, we’ll explore how to find the sum of the first 8 terms of an arithmetic sequence with a clear first term and common difference.", "---", "### What Is an Arithmetic Sequence?", "An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant value, known as the common difference, to the previous term.", "Given:\n- First term ( a = 5 )\n- Common difference ( d = 3 )\n- Number of terms ( n = 8 )", "The sequence looks like:\n5, 8, 11, 14, 17, 20, 23, 26", "---", "### Formula for the Sum of an Arithmetic Sequence", "To find the sum ( S_n ) of the first ( n ) terms of an arithmetic sequence, we use the formula:", "[\nS_n = \frac{n}{2} \ imes (2a + (n - 1)d)\n]", "Alternatively, it can also be written as:", "[\nS_n = \frac{n}{2} \ imes (a + l)\n]", "where:\n- ( a ) is the first term\n- ( l ) is the last term (8th term in this case)\n- ( n ) is the number of terms", "Since we already know ( a ), ( d ), and ( n ), we can use either formula. Here’s the step-by-step calculation using the first formula.", "---", "### Step-by-Step Calculation", "1. Substitute known values into the sum formula:", "[\nS_8 = \frac{8}{2} \ imes (2 \ imes 5 + (8 - 1) \ imes 3)\n]", "2. Simplify inside the parentheses:", "[\nS_8 = 4 \ imes (10 + 7 \ imes 3)\n]\n[\nS_8 = 4 \ imes (10 + 21)\n]\n[\nS_8 = 4 \ imes 31\n]", "3. Multiply:", "[\nS_8 = 124\n]", "---", "### Verification Using the Term-by-Term Addition Method", "For clarity, let’s verify by adding all eight terms:", "[\n5 + 8 + 11 + 14 + 17 + 20 + 23 + 26\n]", "Grouping terms:\n(5 + 26) + (8 + 23) + (11 + 20) + (14 + 17)\n= 31 + 31 + 31 + 31 = 124", "The result confirms our formulaic calculation.", "---", "### Why Knowing This Sum Matters", "Calculating the sum of terms in sequences builds foundational algebra skills. It applies in finance (e.g., calculating total savings over time), physics (like uniform motion), and statistics (averaging sequential data). Mastery of this concept leads to more advanced problem-solving techniques.", "---", "### Conclusion", "The sum of the first 8 terms of the arithmetic sequence starting at 5 with a common difference of 3 is 124. Using the formula ( S_n = \frac{n}{2}(2a + (n-1)d) ) provides an efficient way to compute this, saving time and reducing errors. Practice this method regularly to strengthen your arithmetic sequence skills and build confidence for more complex mathematical challenges.", "---", "Keywords: arithmetic sequence sum, sum of first 8 terms, finding arithmetic series sum, formula arithmetic sequence, step-by-step arithmetic sequence solution, sum formula nth term sequence, algebra practice problems.\nMeta description: Learn how to find the sum of the first 8 terms of an arithmetic sequence with first term 5 and common difference 3 using step-by-step calculation and formula verification. Improve your math skills today!"]
