The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by \(S_n = \frac{n}{2} (2a + (n-1)d)\). Here, \(n = 8\), \(a = 5\), and \(d = 3\).

["# Understanding the Sum of Arithmetic Sequences: A Practical Guide with (S_8)", "Comprehending how to calculate the sum of the first (n) terms of an arithmetic sequence is essential for solving problems in algebra, finance, and data analysis. One of the most widely used formulas for this purpose 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 between consecutive terms,\n- (n) is the number of terms.", "In this article, we’ll explore how to apply this formula step-by-step with a real example: calculating (S_8) for an arithmetic sequence where (a = 5), (d = 3), and (n = 8).", "---", "## What is an Arithmetic Sequence?", "An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the common difference, denoted by (d). The general form of the (n)-th term is:\n[\na_n = a + (n - 1)d\n]", "The sum (S_n) captures how all these terms add together efficiently—without having to manually sum each one.", "---", "## The Formula: (S_n = \frac{n}{2} \left(2a + (n - 1)d\right))", "This formula is derived from pairing terms symmetrically around the middle. It reduces complex addition into a simple computation. Let’s verify it using the values (n = 8), (a = 5), and (d = 3).", "---", "## Step-by-Step Calculation of (S_8)", "### Step 1: Identify the given values\n[\nn = 8, \quad a = 5, \quad d = 3\n]", "### Step 2: Plug into the formula\n[\nS_8 = \frac{8}{2} \left(2 \cdot 5 + (8 - 1) \cdot 3\right)\n]", "### Step 3: Simplify inside the parentheses\nFirst compute each part:\n- (2a = 2 \cdot 5 = 10)\n- ((n - 1)d = 7 \cdot 3 = 21)", "Add them:\n[\n2a + (n - 1)d = 10 + 21 = 31\n]", "### Step 4: Multiply by (\frac{n}{2})\n[\nS_8 = 4 \cdot 31 = 124\n]", "---", "## Why This Formula Matters", "- Efficiency: Instead of calculating (a_1 + a_2 + \cdots + a_8) one by one (which would be tedious), we used a direct algebraic method.\n- Applications: This formula is used in finance (e.g., compound interest over fixed periods), physics (certain motion problems), and computer science (algorithmic complexity).\n- Generalization: The structure reveals how both the first term and the rate of growth interact to determine total accumulation.", "---", "## Conclusion", "Understanding the sum formula (S_n = \frac{n}{2} \left(2a + (n - 1)d\right)) equips you with a powerful tool to analyze and compute cumulative sequences efficiently. In our example with (n=8), (a=5), and (d=3), we calculated:", "[\nS_8 = 124\n]", "This reveals how arithmetic sequences offer not just patterns, but practical methods for quick computation in real-world contexts.", "For further mastery, practice substituting different values and explore related identities—like the closed-form expression involving (n), (a), and (d)—to deepen your grasp of this foundational concept."]