To find the sum of the first five terms of the sequence \(\{a_n\}\) where \(a_n = 2n^2 + 3n + 1\), we calculate each term individually:

["# How to Find the Sum of the First Five Terms of the Sequence (a_n = 2n^2 + 3n + 1)", "Understanding sequences is a fundamental skill in mathematics, and calculating sums of terms efficiently helps strengthen problem-solving abilities. One common task is determining the sum of the first five terms of a defined sequence. In this article, we’ll explore how to find the sum of the first five terms of the sequence given by", "[\na_n = 2n^2 + 3n + 1\n]\nby calculating each term step by step.", "## What Is the Sequence (a_n)?", "The sequence (a_n) is defined by the quadratic expression (2n^2 + 3n + 1), which depends on the term index (n), starting at (n = 1) for the first term. This formula combines a squared term, a linear term, and a constant, making it a polynomial sequence.", "## Step-by-Step Calculation of Each Term", "To find the sum of the first five terms, we substitute (n = 1) through (n = 5) into the formula (a_n = 2n^2 + 3n + 1):", "### Term 1 ((n = 1))\n[\na_1 = 2(1)^2 + 3(1) + 1 = 2 + 3 + 1 = 6\n]", "### Term 2 ((n = 2))\n[\na_2 = 2(2)^2 + 3(2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15\n]", "### Term 3 ((n = 3))\n[\na_3 = 2(3)^2 + 3(3) + 1 = 2(9) + 9 + 1 = 18 + 9 + 1 = 28\n]", "### Term 4 ((n = 4))\n[\na_4 = 2(4)^2 + 3(4) + 1 = 2(16) + 12 + 1 = 32 + 12 + 1 = 45\n]", "### Term 5 ((n = 5))\n[\na_5 = 2(5)^2 + 3(5) + 1 = 2(25) + 15 + 1 = 50 + 15 + 1 = 66\n]", "## Adding the First Five Terms", "Now that we have all five terms—6, 15, 28, 45, and 66—we calculate their sum:", "[\n6 + 15 + 28 + 45 + 66 = 160\n]", "Thus, the sum of the first five terms is 160.", "## Why This Approach Works", "This method relies on direct substitution and arithmetic computation, grounded in algebraic evaluation. By carefully computing each term, we minimize errors and build confidence in handling polynomial expressions. This approach is widely applicable in series summation and algorithmic analysis.", "## Final Thoughts", "Learning to compute sums of sequences like ({a_n} = 2n^2 + 3n + 1) enhances mathematical fluency. Whether for academic purposes, coding challenges, or real-world problem-solving, mastering step-by-step term evaluation is essential. The total sum (S_5 = 160) confirms the accuracy of breaking down each term and adding sequentially.", "Whether you're a student, educator, or enthusiast, mastering these techniques unlocks deeper insights into patterns and structured calculations.", "---\nKeywords: sum of sequence, (a_n = 2n^2 + 3n + 1), first five terms, quadratic sequence, algebra to sum terms, step-by-step calculation."]