A sequence is defined by \( a_n = 3n^2 + 2n + 1 \). Find the sum of the first 5 terms.

["Title: Find the Sum of the First 5 Terms of the Sequence Defined by (a_n = 3n^2 + 2n + 1)", "---", "Introduction\nUnderstanding sequences is fundamental in mathematics, especially when analyzing patterns and computing sums of series. In this article, we explore the sequence defined by the formula (a_n = 3n^2 + 2n + 1) and calculate the sum of its first five terms. This step-by-step approach helps students and learners grasp both the theoretical and practical aspects of sequence summation.", "---", "Understanding the Sequence (a_n = 3n^2 + 2n + 1)\nThe general term (a_n = 3n^2 + 2n + 1) defines a quadratic sequence where each term depends on the square of its position (n), a linear term, and a constant. To uncover the structure of the sequence, we compute the first five terms directly using the formula.", "---", "Calculating the First Five Terms", "Using (a_n = 3n^2 + 2n + 1), compute each term step-by-step:", "- For (n = 1):\n (a_1 = 3(1)^2 + 2(1) + 1 = 3 + 2 + 1 = 6)", "- For (n = 2):\n (a_2 = 3(2)^2 + 2(2) + 1 = 12 + 4 + 1 = 17)", "- For (n = 3):\n (a_3 = 3(3)^2 + 2(3) + 1 = 27 + 6 + 1 = 34)", "- For (n = 4):\n (a_4 = 3(4)^2 + 2(4) + 1 = 48 + 8 + 1 = 57)", "- For (n = 5):\n (a_5 = 3(5)^2 + 2(5) + 1 = 75 + 10 + 1 = 86)", "---", "Summing the First Five Terms", "Now, add (a_1) through (a_5):\n[\nS_5 = a_1 + a_2 + a_3 + a_4 + a_5 = 6 + 17 + 34 + 57 + 86\n]", "Compute step-by-step:\n(6 + 17 = 23)\n(23 + 34 = 57)\n(57 + 57 = 114)\n(114 + 86 = 200)", "Thus, the sum of the first five terms is 200.", "---", "Alternative Academic Insight\nTo deepen understanding, note that the sum of ( \sum_{n=1}^{5} a_n = \sum_{n=1}^{5} (3n^2 + 2n + 1) ) can be expanded using summation properties:\n[\n= 3\sum_{n=1}^{5} n^2 + 2\sum_{n=1}^{5} n + \sum_{n=1}^{5} 1\n]", "Use known summation formulas:\n- ( \sum_{n=1}^{k} n = \frac{k(k+1)}{2} )\n- ( \sum_{n=1}^{k} n^2 = \frac{k(k+1)(2k+1)}{6} )\n- ( \sum_{n=1}^{k} 1 = k )", "Plug in (k = 5):\n- ( \sum n^2 = \frac{5 \cdot 6 \cdot 11}{6} = 55 )\n- ( \sum n = \frac{5 \cdot 6}{2} = 15 )\n- ( \sum 1 = 5 )", "Now compute:\n[\n3(55) + 2(15) + 5 = 165 + 30 + 5 = 200\n]", "This confirms our direct summation and illustrates the power of algebraic formulas.", "---", "Conclusion\nThe sequence defined by (a_n = 3n^2 + 2n + 1) yields clearly definable terms. Calculating and summing the first five terms gives (a_1 = 6), (a_2 = 17), (a_3 = 34), (a_4 = 57), and (a_5 = 86), with total sum:", "[\n6 + 17 + 34 + 57 + 86 = 200\n]", "Understanding how to compute and sum terms in quadratic sequences enhances analytical skills and supports advanced studies in mathematics.", "---", "Keywords: sequence formula, (a_n = 3n^2 + 2n + 1), first five terms, summation, arithmetic sequence, quadratic sequence, math tutorial, sequence sum, algebra example", "Meta description: Learn how to compute the sum of the first 5 terms of the sequence (a_n = 3n^2 + 2n + 1), including step-by-step calculations and summation methods. Perfect for students and math enthusiasts."]