Find the sum of the first 20 terms of the arithmetic sequence where the first term is 5 and the common difference is 3.
["# Finding the Sum of the First 20 Terms of an Arithmetic Sequence", "When studying sequences in mathematics, arithmetic sequences are fundamental due to their predictable patterns. If you're looking to compute the sum of the first 20 terms of a specific arithmetic sequence, understanding the formula and how to apply it efficiently is key. In this article, we’ll walk through step-by-step how to find the sum of the first 20 terms of an arithmetic sequence where the first term is 5 and the common difference is 3.", "---", "## Understanding Arithmetic Sequences", "An arithmetic sequence is a sequence of numbers where each term increases by a constant amount, known as the common difference. The general form of the (n^{th}) term is:", "[\na_n = a_1 + (n - 1)d\n]", "Where:\n- (a_1) is the first term\n- (d) is the common difference\n- (n) is the term number\n- (a_n) is the (n^{th}) term", "---", "## Sum Formula for Arithmetic Sequences", "To find the sum (S_n) of the first (n) terms, we use the formula:", "[\nS_n = \frac{n}{2} \ imes (a_1 + a_n)\n]", "Alternatively, since we may not know the last term directly, we can use:", "[\nS_n = \frac{n}{2} \ imes [2a_1 + (n - 1)d]\n]", "This form is particularly useful when only (a_1) and (d) are given.", "---", "## Applying the Values", "Given:\n- First term ((a_1)) = 5\n- Common difference ((d)) = 3\n- Number of terms ((n)) = 20", "Use the sum formula:", "[\nS_{20} = \frac{20}{2} \ imes [2 \ imes 5 + (20 - 1) \ imes 3]\n]", "Step-by-step:", "1. Calculate (2a_1):\n[\n2 \ imes 5 = 10\n]", "2. Calculate ((n - 1) \ imes d):\n[\n19 \ imes 3 = 57\n]", "3. Add to get (2a_1 + (n - 1)d):\n[\n10 + 57 = 67\n]", "4. Multiply by (\frac{n}{2} = 10):\n[\nS_{20} = 10 \ imes 67 = 670\n]", "---", "## Conclusion", "The sum of the first 20 terms of the arithmetic sequence with (a_1 = 5) and (d = 3) is 670. Understanding how to apply the arithmetic sum formula efficiently can simplify many problems in algebra, finance, or pattern recognition — making mastery of this concept essential for students and math enthusiasts alike.", "---", "Keywords: arithmetic sequence sum, first 20 terms, arithmetic series formula, sum of arithmetic sequence, find sum of 20 terms, common difference 3, first term 5, mathematical sequence sum, formula derivation, algebraic sum calculation", "Meta Description: Learn how to find the sum of the first 20 terms in an arithmetic sequence with (a_1 = 5) and (d = 3) using the standard sum formula. Step-by-step guide and calculation shown.", "---", "📘 Tip: Practice with similar sequences to reinforce your understanding—whether you're solving math problems or exploring real-world applications like cumulative growth or spacing patterns."]
