\( 90 = \frac{n}{2}(10 + (n−1)5) = \frac{n}{2}(5n + 5) = \frac{5n(n+1)}{2} \)

["# Unlocking the Sum: Mastering the Average Formula for an Arithmetic Sequence", "Understanding how to calculate the sum of an arithmetic sequence is a fundamental skill in mathematics, especially in algebra and pre-calculus. One frequently encountered formula expresses the sum ( S_n ) of the first ( n ) terms of an arithmetic sequence defined as:", "[\n90 = \frac{n}{2}(10 + (n - 1) \cdot 5)\n]\n[\n= \frac{n}{2}(5n + 5)\n]\n[\n= \frac{5n(n + 1)}{2}\n]", "This elegant formula reveals how the sum depends on the number of terms and the average of the first and last terms. In this article, we’ll explore this formula step-by-step, simplify it, and explain its real-world applications.", "## What Is an Arithmetic Sequence?", "An arithmetic sequence is a sequence of numbers where each term increases by a constant difference, called the common difference ( d ). For example: 10, 15, 20, ..., where ( d = 5 ).", "If ( a_1 ) is the first term and ( n ) the number of terms, the sum of the first ( n ) terms ( S_n ) is given by:", "[\nS_n = \frac{n}{2}(a_1 + a_n)\n]", "Because the sequence increases uniformly, ( a_n = a_1 + (n - 1)d ).", "## Breaking Down the Formula", "Given:\n[\nS_n = \frac{n}{2}(10 + (n - 1) \cdot 5)\n]", "Here, ( a_1 = 10 ), ( a_n = 10 + (n - 1) \cdot 5 ), and ( d = 5 ).", "Let’s simplify the expression step-by-step:", "1. Start with:\n[\n\frac{n}{2}(10 + 5(n - 1))\n]", "2. Distribute inside the parentheses:\n[\n10 + 5n - 5 = 5n + 5\n]", "3. Substitute back:\n[\n\frac{n}{2}(5n + 5)\n]", "4. Factor out 5:\n[\n\frac{5n}{2}(n + 1)\n]", "So the sum formula becomes:\n[\nS_n = \frac{5n(n + 1)}{2}\n]", "This elegant simplification shows that the sum depends on ( n ) and the formula behaves like the sum of the first ( n ) natural numbers scaled by 5.", "## Why This Formula Matters", "The formula ( S_n = \frac{n}{2}(a_1 + a_n) ) unlocks powerful problem-solving techniques:", "- Efficient Summation: Instead of adding each term, use a direct formula—especially useful for large ( n ).\n- Real-World Applications: Calculating total distance traveled under constant acceleration (modeling motion), evaluating cumulative payments, or forecasting linear growth.\n- Bridge to Calculus & Series: Forms the foundation for summing arithmetic series, used in integrals and financial mathematics.", "## Applying the Formula in Practice", "Suppose you invest $10 monthly with a fixed 5% increase each term. After 5 months, the total investment sum follows:", "[\nS_5 = \frac{5 \cdot 5 \cdot (5 + 1)}{2} = \frac{5 \cdot 5 \cdot 6}{2} = 75\n]", "This means after 5 terms, your cumulative contribution is $75—modeling real financial growth simply.", "## Final Thoughts", "Mastering the sum formula for arithmetic sequences transforms how you approach linear growth problems. Recognizing patterns, simplifying expressions, and applying formulas strategically sharpen mathematical intuition and empower practical problem-solving across science, finance, and engineering.", "Whether calculating averages, projections, or cumulative trends—remember: sum = average × number of terms, and in arithmetic sequences, this average reveals a clean structure through elegant formulas like ( \frac{n}{2}(10 + 5(n-1)) ), simplified to ( \frac{5n(n+1)}{2} ).", "---", "Keywords: arithmetic sequence sum formula, sum of arithmetic series, linear growth formula, average of sequence, mathematical summation techniques, Fibonacci-like sum patterns, algebra simplification."]