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

["Understanding the Sequence: (a_n = 3n^2 + 2n + 1) and the Sum of the First 5 Terms", "In mathematics, sequences play a fundamental role in understanding patterns and formulating formulas. One such sequence is defined by the quadratic expression:", "[\na_n = 3n^2 + 2n + 1\n]", "This formula generates each term based on its position (n) in the sequence. In this article, we’ll explore how to compute the first five terms and find their sum—essential knowledge for students, educators, and enthusiasts interested in sequences and series.", "### Step 1: Calculate Each of the First 5 Terms", "Using the formula (a_n = 3n^2 + 2n + 1), we plug in (n = 1) through (n = 5):", "- For (n = 1):\n[\na_1 = 3(1)^2 + 2(1) + 1 = 3 + 2 + 1 = 6\n]", "- For (n = 2):\n[\na_2 = 3(2)^2 + 2(2) + 1 = 12 + 4 + 1 = 17\n]", "- For (n = 3):\n[\na_3 = 3(3)^2 + 2(3) + 1 = 27 + 6 + 1 = 34\n]", "- For (n = 4):\n[\na_4 = 3(4)^2 + 2(4) + 1 = 48 + 8 + 1 = 57\n]", "- For (n = 5):\n[\na_5 = 3(5)^2 + 2(5) + 1 = 75 + 10 + 1 = 86\n]", "So, the first five terms are:\n6, 17, 34, 57, and 86.", "### Step 2: Sum the First 5 Terms", "Now, compute the total sum:", "[\nS_5 = a_1 + a_2 + a_3 + a_4 + a_5 = 6 + 17 + 34 + 57 + 86\n]", "Adding step by step:", "[\n6 + 17 = 23\n]\n[\n23 + 34 = 57\n]\n[\n57 + 57 = 114\n]\n[\n114 + 86 = 200\n]", "Thus, the sum of the first five terms is:", "[\nS_5 = 200\n]", "### Why Equal This Summation Matters", "Understanding the sum of a quadratic sequence like (a_n = 3n^2 + 2n + 1) is valuable in various applications, including:", "- Series Analysis: Computing partial sums efficiently.\n- Programming & Algorithms: Modeling performance or cumulative growth.\n- Finance & Economics: Calculating cumulative returns or costs over periods.\n- Data Science: Analyzing trends expressed linearly or quadratically in time series.", "### Bonus: General Formula for the Sum", "For deeper study, the sum of the first (n) terms of (a_n = 3n^2 + 2n + 1) can be derived using standard summation formulas:", "[\n\sum_{k=1}^{n} (3k^2 + 2k + 1) = 3\sum_{k=1}^{n} k^2 + 2\sum_{k=1}^{n} k + \sum_{k=1}^{n} 1\n]", "Using:", "[\n\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6},\quad\n\sum_{k=1}^{n} k = \frac{n(n+1)}{2},\quad\n\sum_{k=1}^{n} 1 = n\n]", "Substitute and simplify:", "[\nS_n = 3\left(\frac{n(n+1)(2n+1)}{6}\right) + 2\left(\frac{n(n+1)}{2}\right) + n = \frac{n(n+1)(2n+1)}{2} + n(n+1) + n\n]", "This simplifies to:", "[\nS_n = \frac{n(n+1)(2n+1 + 2) }{2} + n = \frac{n(n+1)(2n+3)}{2} + n\n]", "For (n = 5), plugging in confirms our earlier result:", "[\nS_5 = \frac{5 \cdot 6 \cdot 13}{2} + 5 = \frac{390}{2} + 5 = 195 + 5 = 200\n]", "---", "Conclusion", "The sequence defined by (a_n = 3n^2 + 2n + 1) produces terms defined by a quadratic rule. Calculating and summing the first five terms directly or by general summation formulas reveals a total of 200.", "Whether you’re solving for sums in algebra, supporting computational tasks, or modeling real-world growth, mastering such sequences equips you with powerful analytical tools.", "---", "Keywords:\nsequence defined by (a_n = 3n^2 + 2n + 1), sum of sequence, quadratic sequence, find (a_1 + a_2 + a_3 + a_4 + a_5), summation formulas, mathematical education, pattern recognition.", "---", "Stay tuned for deeper explorations into sequences, series convergence, and their real-life applications!"]