A sequence is defined by \(a_n = 2n^2 + 3n + 1\). What is the sum of the first 5 terms of this sequence?
["Understanding the Sequence Defined by (a_n = 2n^2 + 3n + 1): Sum of the First 5 Terms", "The mathematical study of sequences is fundamental in algebra and beyond. One interesting type of sequence is defined explicitly by a formula, such as (a_n = 2n^2 + 3n + 1). In this article, we explore the sequence, derive its first five terms, and calculate their sum—essential knowledge for students, educators, and math enthusiasts.", "### What Is a Sequence?", "A sequence is a list of numbers generated by a consistent rule or formula. When defined explicitly as (a_n = 2n^2 + 3n + 1), each term corresponds to a natural number index (n). Unlike recursive sequences, this formula lets us compute any term directly without needing prior values.", "### Calculating the First 5 Terms", "Let’s compute the first five terms by plugging (n = 1) to (n = 5) into the formula (a_n = 2n^2 + 3n + 1):", "- For (n = 1):\n (a_1 = 2(1)^2 + 3(1) + 1 = 2 + 3 + 1 = 6)", "- For (n = 2):\n (a_2 = 2(2)^2 + 3(2) + 1 = 8 + 6 + 1 = 15)", "- For (n = 3):\n (a_3 = 2(3)^2 + 3(3) + 1 = 18 + 9 + 1 = 28)", "- For (n = 4):\n (a_4 = 2(4)^2 + 3(4) + 1 = 32 + 12 + 1 = 45)", "- For (n = 5):\n (a_5 = 2(5)^2 + 3(5) + 1 = 50 + 15 + 1 = 66)", "The first five terms are: 6, 15, 28, 45, 66.", "### Summing the First 5 Terms", "Adding these values gives:", "[\n6 + 15 + 28 + 45 + 66 = 160\n]", "Alternatively, summing step-by-step:\n(6 + 15 = 21)\n(21 + 28 = 49)\n(49 + 45 = 94)\n(94 + 66 = 160)", "### Why Sum Sequences?", "Summing terms in a sequence helps evaluate totals in real-world applications, such as modeling growth, financial projections, or algorithmic complexity. Understanding expressions like (a_n = 2n^2 + 3n + 1) allows efficient computation and pattern recognition.", "### Conclusion", "The sequence defined by (a_n = 2n^2 + 3n + 1) produces terms that grow quadratically. The sum of the first five terms—6, 15, 28, 45, and 66—is 160. Mastering such sequences enhances analytical skills and supports deeper mathematical reasoning.", "Whether you're a learner or a teacher, recognizing how to compute and sum sequence terms empowers you to solve complex problems with precision.", "---", "Key Takeaways:", "- The explicit formula (a_n = 2n^2 + 3n + 1) defines a quadratic sequence.\n- Calculating the first five terms involves substituting (n = 1) to (5).\n- Summing: (6 + 15 + 28 + 45 + 66 = 160).\n- Summing sequences improves analytical and computational skills.", "Keywords: sequence defined by (a_n = 2n^2 + 3n + 1), sum of first 5 terms, explicit formula, quadratic sequence, mathematics education, algebra, sequence sum, learn math sequences."]
