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

["Understanding Sequences: How to Compute the Sum of the First 5 Terms of ( a_n = 3n^2 - 2n + 1 )", "When working with mathematical sequences, one common task is to find the sum of the first ( n ) terms. In this article, we explore how to determine the sum of the first 5 terms of the sequence defined by the formula:", "[\na_n = 3n^2 - 2n + 1\n]", "This sequence models a quadratic relation where each term depends on the square of its index ( n ). Understanding how to compute cumulative sums for such expressions is essential in mathematics, physics, engineering, and data analysis.", "---", "### Step 1: Understand the Sequence Formula", "The sequence is defined explicitly by:\n[\na_n = 3n^2 - 2n + 1\n]", "This means the ( n )-th term in the sequence is generated by plugging in integer values of ( n ), starting at 1.", "---", "### Step 2: Compute the First 5 Terms", "To find the sum of the first 5 terms, we evaluate ( a_n ) for ( n = 1 ) through ( n = 5 ):", "- For ( n = 1 ):\n [\n a_1 = 3(1)^2 - 2(1) + 1 = 3 - 2 + 1 = 2\n ]", "- For ( n = 2 ):\n [\n a_2 = 3(2)^2 - 2(2) + 1 = 12 - 4 + 1 = 9\n ]", "- For ( n = 3 ):\n [\n a_3 = 3(3)^2 - 2(3) + 1 = 27 - 6 + 1 = 22\n ]", "- For ( n = 4 ):\n [\n a_4 = 3(4)^2 - 2(4) + 1 = 48 - 8 + 1 = 41\n ]", "- For ( n = 5 ):\n [\n a_5 = 3(5)^2 - 2(5) + 1 = 75 - 10 + 1 = 66\n ]", "So the first five terms are:\n[\n2,\ 9,\ 22,\ 41,\ 66\n]", "---", "### Step 3: Sum the Terms", "Now sum these values:", "[\n2 + 9 + 22 + 41 + 66\n]", "Add step-by-step:\n- ( 2 + 9 = 11 )\n- ( 11 + 22 = 33 )\n- ( 33 + 41 = 74 )\n- ( 74 + 66 = 140 )", "Thus, the sum of the first 5 terms is:", "[\n\sum_{n=1}^{5} a_n = 140\n]", "---", "### Alternative: Use Summation Formulas", "For faster computation—especially useful for larger ( n )—we can use known summation formulas.", "Given ( a_n = 3n^2 - 2n + 1 ), the sum of the first 5 terms is:", "[\n\sum_{n=1}^{5} a_n = 3\sum_{n=1}^{5} n^2 - 2\sum_{n=1}^{5} n + \sum_{n=1}^{5} 1\n]", "We use the standard formulas:\n- ( \sum_{n=1}^{k} n^2 = \dfrac{k(k+1)(2k+1)}{6} )\n- ( \sum_{n=1}^{k} n = \dfrac{k(k+1)}{2} )\n- ( \sum_{n=1}^{k} 1 = k )", "For ( k = 5 ):", "- ( \sum n^2 = \dfrac{5 \cdot 6 \cdot 11}{6} = 55 )\n- ( \sum n = \dfrac{5 \cdot 6}{2} = 15 )\n- ( \sum 1 = 5 )", "Now substitute:", "[\n3(55) - 2(15) + 5 = 165 - 30 + 5 = 140\n]", "This confirms our earlier result using two different methods.", "---", "### Conclusion", "The sum of the first 5 terms of the sequence defined by ( a_n = 3n^2 - 2n + 1 ) is 140. Whether calculating manually or using summation formulas, consistency ensures accuracy. Understanding sequences and their summation not only improves problem-solving skills but also supports applications across scientific computing and algorithm design.", "For future learners, practicing such expressions helps build intuition for more complex recursive and closed-form summations.", "---", "Keywords: sequence sum, explicit formula sum, quadratic sequence, summation of sequences, ( a_n = 3n^2 - 2n + 1 ), mathematical summation, algebra practice problem, study math sequences.", "Meta Description: Learn how to find the sum of the first 5 terms of the sequence defined by ( a_n = 3n^2 - 2n + 1 ) using direct computation and summation formulas. Step-by-step explanation with verification."]