Solution: We compute $ \sum_{n=1}^{10} P(n) = \sum_{n=1}^{10} (n^2 + 3n + 5) = \sum_{n=1}^{10} n^2 + 3\sum_{n=1}^{10} n + \sum_{n=1}^{10} 5 $.
["Title: Breakdown & Calculation of the Summation: Efficiently Computing $\sum_{n=1}^{10} (n^2 + 3n + 5)$", "When faced with summation problems, especially those involving polynomials like $ \sum_{n=1}^{10} (n^2 + 3n + 5) $, a strategic approach ensures clarity, accuracy, and computational efficiency. This article walks through the step-by-step computation of the sum using proven mathematical techniques, leveraging known summation formulas to simplify the calculation of $ \sum_{n=1}^{10} (n^2 + 3n + 5) $.", "---", "### Understanding the Problem", "We aim to compute:", "$$\n\sum_{n=1}^{10} (n^2 + 3n + 5)\n$$", "By the distributive property of summation, this expression expands to:", "$$\n\sum_{n=1}^{10} n^2 + 3 \sum_{n=1}^{10} n + \sum_{n=1}^{10} 5\n$$", "This decomposition transforms a seemingly complex sum into three manageable parts using well-known summation formulas.", "---", "### Step 1: Compute $ \sum_{n=1}^{10} n^2 $", "The formula for the sum of squares of the first $k$ natural numbers is:", "$$\n\sum_{n=1}^{k} n^2 = \frac{k(k+1)(2k+1)}{6}\n$$", "For $k = 10$:", "$$\n\sum_{n=1}^{10} n^2 = \frac{10 \cdot 11 \cdot 21}{6} = \frac{2310}{6} = 385\n$$", "---", "### Step 2: Compute $ \sum_{n=1}^{10} n $", "The sum of the first $k$ natural numbers is:", "$$\n\sum_{n=1}^{k} n = \frac{k(k+1)}{2}\n$$", "For $k = 10$:", "$$\n\sum_{n=1}^{10} n = \frac{10 \cdot 11}{2} = 55\n$$", "Multiply this by 3 as in our original expression:", "$$\n3 \sum_{n=1}^{10} n = 3 \cdot 55 = 165\n$$", "---", "### Step 3: Compute $ \sum_{n=1}^{10} 5 $", "Since 5 is constant across all values of $n$, summing it 10 times is straightforward:", "$$\n\sum_{n=1}^{10} 5 = 5 \cdot 10 = 50\n$$", "---", "### Final Calculation", "Adding all three components together:", "$$\n\sum_{n=1}^{10} (n^2 + 3n + 5) = 385 + 165 + 50 = \boxed{600}\n$$", "---", "### Why This Approach Is Efficient", "By breaking the sum into component parts, we avoid manually computing each term individually, reducing computational complexity and minimizing errors. Utilizing standard summation formulas ensures speed and accuracy, especially useful when dealing with larger ranges (e.g., up to $k = 1000$).", "This method is a cornerstone technique in algebra, discrete mathematics, and algorithmic analysis, proving invaluable in both textbook exercises and real-world applications like performance analysis and financial modeling.", "---", "Conclusion\nUnderstanding how to decompose and compute polynomial sums using summation formulas enhances problem-solving efficiency. The total value of $ \sum_{n=1}^{10} (n^2 + 3n + 5) $ is 600, derived cleanly by leveraging well-established mathematical identities.", "---", "Keywords: summation formula, $ \sum n^2 $, $ \sum n $, $ \sum c $, $ n = 1 $ to $ 10 $, algebra techniques, computational math, discrete summation, mathematical summing, educational calculation."]
