["Evaluate the Sum: A Step-by-Step Guide to Computing (\sum_{k=1}^{5} (2k + 1))", "Understanding how to evaluate summations is a fundamental skill in mathematics, especially in algebra and discrete mathematics. This article provides a clear, structured evaluation of the sum (\sum_{k=1}^{5} (2k + 1)), explaining each step to help learners master this concept.", "---", "### Step 1: Understand What the Sum Means", "The notation
\n[
\n\sum_{k=1}^{5} (2k + 1)
\n]
\nmeans that we are summing the expression (2k + 1) for each integer value of (k) from 1 to 5, inclusive.", "In other words, compute:", "[
\n(2 \cdot 1 + 1) + (2 \cdot 2 + 1) + (2 \cdot 3 + 1) + (2 \cdot 4 + 1) + (2 \cdot 5 + 1)
\n]", "---", "### Step 2: Compute Each Term Individually", "Let’s evaluate each term of the sum:", "- For (k = 1): (2(1) + 1 = 2 + 1 = 3)
\n- For (k = 2): (2(2) + 1 = 4 + 1 = 5)
\n- For (k = 3): (2(3) + 1 = 6 + 1 = 7)
\n- For (k = 4): (2(4) + 1 = 8 + 1 = 9)
\n- For (k = 5): (2(5) + 1 = 10 + 1 = 11)", "---", "### Step 3: Add All Terms Together", "Now, perform the addition:
\n[
\n3 + 5 + 7 + 9 + 11
\n]", "Group the terms for easier calculation:
\n[
\n(3 + 11) + (5 + 9) + 7 = 14 + 14 + 7 = 35
\n]", "Alternatively, summing step-by-step:
\n[
\n3 + 5 = 8
\n8 + 7 = 15
\n15 + 9 = 24
\n24 + 11 = 35
\n]", "---", "### Step 4: Final Result", "Thus,
\n[
\n\sum_{k=1}^{5} (2k + 1) = 35
\n]", "---", "### Alternative Algebraic Method: Simplify Before Summing", "For more efficiency, especially with arithmetic sequences, use summation formulas.", "Note:
\n[
\n\sum_{k=1}^{n} (2k + 1) = 2\sum_{k=1}^{n} k + \sum_{k=1}^{n} 1 = 2 \cdot \frac{n(n+1)}{2} + n = n(n+1) + n = n(n+2)
\n]", "Here, (n = 5), so:
\n[
\n5(5 + 2) = 5 \cdot 7 = 35
\n]", "---", "### Why This Sum Matters", "This type of summation is common in problems involving arithmetic sequences, area calculations, and sequence modeling. Mastering such sums helps students build a strong foundation for calculus, discrete math, and algorithm analysis.", "---", "### Conclusion", "The sum (\sum_{k=1}^{5} (2k + 1)) evaluates to 35 through direct computation or algebraic simplification. Whether you integrate each term or use summation formulas, understanding these methods strengthens your mathematical toolkit.", "---", "Keywords: evaluate sum, (\sum_{k=1}^{5} (2k + 1)), arithmetic sequence sum, step-by-step evaluation, algebra tutorial, summation techniques.
\nMeta Description: Learn how to compute and evaluate the sum (\sum_{k=1}^{5} (2k + 1)) using direct calculation and algebraic formulas. Find step-by-step solutions and formulas for future reference."]
